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# R.D. Sharma solutions for Class 12 Mathematics chapter 24 - Scalar Or Dot Product

## Chapter 24 - Scalar Or Dot Product

#### Page 4

Q 1 | Page 4

Represent the following graphically:
(i) a displacement of 40 km, 30° east of north
(ii) a displacement of 50 km south-east
(iii) a displacement of 70 km, 40° north of west.

Q 2 | Page 4

Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec2

Q 3 | Page 4

Classify the following as scalars and vector quantities:
(i) Time period
(ii) Distance
(iii) displacement
(iv) Force
(v) Work
(vi) Velocity
(vii) Acceleration

Q 5.1 | Page 4

Answer the following as true or false:
$\vec{a}$ and $\vec{a}$  are collinear.

True

False

Q 5.2 | Page 4

Answer the following as true or false:
Two collinear vectors are always equal in magnitude.

true

False

Q 5.3 | Page 4

Answer the following as true or false:
Zero vector is unique.

true

false

Q 5.4 | Page 4

Answer the following as true or false:
Two vectors having same magnitude are collinear.

true

false

Q 5.5 | Page 4

Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.

true

false

#### Page 17

Q 1 | Page 17

If P, Q and R are three collinear points such that $\vec{PQ} = \vec{a}$ and $\vec{PQ} = \vec{a}$. Find the vector $\vec{PR}$

Q 2 | Page 17

Give a condition that three vectors $\vec{a}$, $\vec{b}$ and $\vec{b}$  form the three sides of a triangle. What are the other possibilities?

Q 3 | Page 17

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{a}$ + $\vec{b}$  and $\vec{a}$ − $\vec{b}$.

Q 4 | Page 17

If $\vec{a}$ is a vector and m is a scalar such that m $\vec{a}$ = $\vec{0}$, then what are the alternatives for m and $\vec{a}$ ?

Q 5.1 | Page 17

If $\vec{a,} \vec{b}$ are two vectors, then write the truth value of the following statement:
$\vec{a} = - \vec{b} \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|$

Q 5.2 | Page 17

If $\vec{a,} \vec{b}$ are two vectors, then write the truth value of the following statement:
$\vec{a} = - \vec{b} \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right|$

Q 5.3 | Page 17

If $\vec{a,} \vec{b}$ are two vectors, then write the truth value of the following statement:
$\left| \vec{a} \right| = \left| \vec{b} \right| \Rightarrow \vec{a} = \vec{b}$

Q 6 | Page 17

ABCD is a quadrilateral. Find the sum the vectors $\vec{BA} , \vec{BC} , \vec{CD}$ and $\vec{DA}$

Q 7.1 | Page 17

ABCDE is a pentagon, prove that
$\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} + \vec{EA} = \vec{0}$

Q 7.2 | Page 17

ABCDE is a pentagon, prove that
$\vec{AB} + \vec{AE} + \vec{BC} + \vec{DC} + \vec{ED} + \vec{AC} = 3 \vec{AC}$

Q 8 | Page 17

Prove that the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector.

Q 9 | Page 17

If P is a point and ABCD is a quadrilateral and $\vec{AP} + \vec{PB} + \vec{PD} = \vec{PC}$, show that ABCD is a parallelogram.

Q 10 | Page 17

Five forces $\vec{AB,} \vec{AC,} \vec{AD,} \vec{AE}$ and $\vec{AF}$ act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 $\vec{AO,}$ where O is the centre of hexagon.

#### Pages 23 - 24

Q 1 | Page 23

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$, respectively in the ratio 1 : 2 internally and externally.

Q 2 | Page 24

Let $\vec{a,} \vec{b,} \vec{c,} \vec{d}$ be the position vectors of the four distinct points ABCD. If $\vec{b} - \vec{a} = \vec{c} - \vec{d}$, then show that ABCD is a parallelogram.

Q 3 | Page 24

If $\vec{a,} \vec{b}$ are the position vectors of A, B respectively, find the position vector of a point C in AB produced such that AC = 3 AB and that a point D in BA produced such that BD = 2BA.

Q 4 | Page 24

Show that the four points A, B, C, D with position vectors $\vec{a,} \vec{b,} \vec{c,} \vec{d}$ respectively such that $3 \vec{a} - 2 \vec{b} + 5 \vec{c} - 6 \vec{d} = 0,$ are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.

Q 5 | Page 24

Show that the four points P, Q, R, S with position vectors $\vec{p}$, $\vec{q}$, $\vec{r}$, $\vec{s}$ respectively such that 5 $\vec{p}$ − 2 $\vec{q}$ + 6 $\vec{r}$ − 9 $\vec{s}$ $\vec{0}$, are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.

Q 6 | Page 24

The vertices A, B, C of triangle ABC have respectively position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$  with respect to a given origin O. Show that the point D where the bisector of ∠ A meets BC has position vector $\vec{d} = \frac{\beta \vec{b} + \gamma \vec{c}}{\beta + \gamma},\text{ where }\beta = \left| \vec{c} - \vec{a} \right| \text{ and, }\gamma = \left| \vec{a} - \vec{b} \right|$
Hence, deduce that the incentre I has position vector
$\frac{\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c}}{\alpha + \beta + \gamma},\text{ where }\alpha = \left| \vec{b} - \vec{c} \right|$

#### Pages 36 - 37

Q 1 | Page 36

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that $\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}$

Q 2 | Page 36

Show that the sum of three vectors determined by the medians of a triangle directed from the vertices is zero.

Q 3 | Page 37

ABCD is a parallelogram and P is the point of intersection of its diagonals. If O is the origin of reference, show that
$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4 \vec{OP}$

Q 4 | Page 37

Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.

Q 5 | Page 37

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD; BC and AD. Show that$\vec{PA} + \vec{PB} + \vec{PC} + \vec{PD} = 4 \vec{PQ}$, where P is any point.

Q 6 | Page 37

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.

#### Pages 42 - 43

Q 1 | Page 42

If the position vector of a point (−4, −3) be $\vec{a,}$ find $\left| \vec{a} \right|$

Q 2 | Page 42

If the position vector $\vec{a}$ of a point (12, n) is such that $\left| \vec{a} \right|$ = 13, find the value (s) of n.

Q 3 | Page 42

Find a vector of magnitude 4 units which is parallel to the vector $\sqrt{3} \hat{i} + \hat{j}$

Q 4.1 | Page 42

Express $\vec{AB}$  in terms of unit vectors $\hat{i}$ and $\hat{j}$, when the points are A (4, −1), B (1, 3)
Find $\left| \vec{A} B \right|$ in each case.

Q 4.2 | Page 42

Express $\vec{AB}$  in terms of unit vectors $\hat{i}$ and $\hat{j}$, when the points are A (−6, 3), B (−2, −5)
Find $\left| \vec{A} B \right|$ in each case.

Q 5 | Page 42

Find the coordinates of the tip of the position vector which is equivalent to $\vec{A} B$, where the coordinates of A and B are (−1, 3) and (−2, 1) respectively.

Q 6 | Page 43

ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.

Q 7 | Page 43

If the position vectors of the points A (3, 4), B (5, −6) and C (4, −1) are $\vec{a,}$ $\vec{b,}$ $\vec{c}$ respectively, compute $\vec{a} + 2 \vec{b} - 3 \vec{c}$.

Q 8 | Page 43

If $\vec{a}$ be the position vector whose tip is (5, −3), find the coordinates of a point B such that $\vec{AB} =$ $\vec{a}$, the coordinates of A being (4, −1).

Q 9 | Page 43

Show that the points 2 $\hat{i}, - \hat{i}-4$
$\hat{j}$ and $\hat{i}+4$  form an isosceles triangle.

Q 10 | Page 43

Find a unit vector parallel to the vector $\hat{i} + \sqrt{3} \hat{j}$

Q 11 | Page 43

The position vectors of points A, B and C  are $\lambda \hat{i} +$ 3 $\hat{j}$,12$\hat{i} + \mu$ $\hat{j}$ and 11$\hat{i} -$ 3 $\hat{j}$ respectively. If C divides the line segment joining and B in the ratio 3:1, find the values of $\lambda$ and $\lambda$

#### Pages 48 - 49

Q 1 | Page 48

Find the magnitude of the vector $\vec{a} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k} .$

Q 2 | Page 48

Find the unit vector in the direction of $3 \hat{i} + 4 \hat{j} - 12 \hat{k} .$

Q 3 | Page 48

Find a unit vector in the direction of the resultant of the vectors
$\hat{i} - \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} - 2 \hat{k} \text{ and }\hat{i} + 2 \hat{j} - 2 \hat{k} .$

Q 4 | Page 49

The adjacent sides of a parallelogram are represented by the vectors $\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .$
Find unit vectors parallel to the diagonals of the parallelogram.

Q 5 | Page 48
$\text{If }\vec{a} = 3 \hat{i} - \hat{j} - 4 \hat{k} , \vec{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}\text{ and }\vec{c} = \hat{i} + 2 \hat{j} - \hat{k} ,\text{ find }\left| 3 \vec{a} - 2 \vec{b} + 4 \vec{c} \right| .$

Q 6 | Page 49

If $\vec{PQ} = 3 \hat{i} + 2 \hat{j} - \hat{k}$ and the coordinates of P are (1, −1, 2), find the coordinates of Q.

Q 7 | Page 49

Prove that the points $\hat{i} - \hat{j} , 4 \hat{i} + 3 \hat{j} + \hat{k} \text{ and }2 \hat{i} - 4 \hat{j} + 5 \hat{k}$ are the vertices of a right-angled triangle.

Q 8 | Page 49

If the vertices of a triangle are the points with position vectors $a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} , c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,$
what are the vectors determined by its sides? Find the length of these vectors.

Q 9 | Page 49

Find the vector from the origin O to the centroid of the triangle whose vertices are (1, −1, 2), (2, 1, 3) and (−1, 2, −1).

Q 10.1 | Page 49

Find the position vector of a point R which divides the line segment joining points $P \left( \hat{i} + 2 \hat{j} + \hat{k} \right) \text{ and Q }\left( - \hat{i} + \hat{j} + \hat{k} \right)$ internally.

Q 10.2 | Page 49

Find the position vector of the mid-point of the vector joining the points $P \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\text{ and Q }\left( 4 \hat{i} + \hat{j} - 2 \hat{k} \right) .$

Q 11 | Page 49

Find the position vector of the mid-point of the vector joining the points $P \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\text{ and Q }\left( 4 \hat{i} + \hat{j} - 2 \hat{k} \right) .$

Q 12 | Page 49

Find the unit vector in the direction of vector $\vec{PQ} ,$

where P and Q are the points (1, 2, 3) and (4, 5, 6).

Q 13 | Page 49

Show that the points $A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)$ are the vertices of a right angled triangle.

Q 14 | Page 49

Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, −2).

Q 15 | Page 49

Find the value of x for which $x \left( \hat{i} + \hat{j} + \hat{k} \right)$ is a unit vector.

Q 16 | Page 49

If $\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,$ find a unit vector parallel to $2 \vec{a} - \vec{b} + 3 \vec{c .}$

Q 17 | Page 49

If $\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 4 \hat{i} - 2 \hat{j} + 3 \hat{k} and \vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,$ find a vector of magnitude 6 units which is parallel to the vector $2 \vec{a} - \vec{b} + 3 \vec{c .}$

Q 18 | Page 49

Find a vector of magnitude of 5 units parallel to the resultant of the vectors $\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} +\widehat{k} .$

Q 19 | Page 49

The two vectors $\hat{j} + \hat{k}$ and $3 \hat{i} - \hat{j} + 4 \hat{k}$ represents the sides $\vec{AB}$ and $\vec{AC}$ respectively of a triangle ABC. Find the length of the median through A.

#### Pages 60 - 61

Q 1 | Page 60

Show that the points A, B, C with position vectors $\vec{a} - 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + 3 \vec{b} - 4 \vec{c}$ and $- 7 \vec{b} + 10 \vec{c}$ are collinear.

Q 2.1 | Page 60

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are non-coplanar vectors, prove that the points having the following position vectors are collinear: $\vec{a,} \vec{b,} 3 \vec{a} - 2 \vec{b}$

Q 2.2 | Page 60

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are non-coplanar vectors, prove that the points having the following position vectors are collinear: $\vec{a} + \vec{b} + \vec{c} , 4 \vec{a} + 3 \vec{b} , 10 \vec{a} + 7 \vec{b} - 2 \vec{c}$

Q 3 | Page 60

Prove that the points having position vectors $\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{i} - 5 \hat{k}$ are collinear.

Q 4 | Page 60

If the points with position vectors $10 \hat{i} + 3 \hat{j} , 12 \hat{i} - 5 \hat{j}\text{ and a }\hat{i} + 11 \hat{j}$ are collinear, find the value of a.

Q 5 | Page 61

If $\vec{a,} \vec{b}$ are two non-collinear vectors, prove that the points with position vectors $\vec{a,} \vec{b}$ are two non-collinear vectors, prove that the points with position vectors $\vec{a} + \vec{b,} \vec{a} - \vec{b}\text{ and }\vec{a} + \lambda \vec{b}$ are collinear for all real values of λ.

Q 6 | Page 61

If $\vec{AO} + \vec{OB} = \vec{BO} + \vec{OC} ,$ prove that A, B, C are collinear points.

Q 7 | Page 61

Show that the vectors $2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}$ are collinear.

Q 8 | Page 61

If the points A(m, −1), B(2, 1) and C(4, 5) are collinear, find the value of m.

Q 9 | Page 61

Show that the points (3, 4), (−5, 16) and (5, 1) are collinear.

Q 10 | Page 61

If the vectors $\vec{a} = 2 \hat{i} - 3 \hat{j}$ and $\vec{b} = - 6 \hat{i} + m \hat{j}$ are collinear, find the value of m.

Q 11 | Page 61

Show that the points A (1, −2, −8), B (5, 0, −2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Q 12 | Page 61

Using vectors show that the points A (−2, 3, 5), B (7, 0, −1) C (−3, −2, −5) and D (3, 4, 7) are such that AB and CD intersect at the point P (1, 2, 3).

Q 13 | Page 61

Using vectors, find the value of λ such that the points (λ, −10, 3), (1, −1, 3) and (3, 5, 3) are collinear.

#### Pages 65 - 66

Q 1.1 | Page 65

Show that the points whose position vectors are as given below are collinear:
$2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k} \text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}$

Q 1.2 | Page 65

Show that the points whose position vectors are as given below are collinear: $3 \hat{i} - 2 \hat{j} + 4 \hat{k}, \hat{i} + \hat{j} + \hat{k}\text{ and }- \hat{i} + 4 \hat{j} - 2 \hat{k}$

Q 2.1 | Page 65

Using vector method, prove that the following points are collinear:
A (6, −7, −1), B (2, −3, 1) and C (4, −5, 0)

Q 2.2 | Page 65

Using vector method, prove that the following points are collinear:
A (2, −1, 3), B (4, 3, 1) and C (3, 1, 2)

Q 2.3 | Page 65

Using vector method, prove that the following points are collinear:
A (1, 2, 7), B (2, 6, 3) and C (3, 10, −1)

Q 2.4 | Page 65

Using vector method, prove that the following points are collinear:
A (−3, −2, −5), B (1, 2, 3) and C (3, 4, 7)

Q 3 | Page 65

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(1) $5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c}\text{ and }3 \vec{a} + 20 \vec{b} + 5 \vec{c}$

(2) $\vec{a} - 2 \vec{b} + 3 \vec{c} , - 3 \vec{b} + 5 \vec{c}\text{ and }- 2 \vec{a} + 3 \vec{b} - 4 \vec{c}$
Q 4 | Page 65

Show that the four points having position vectors
$6 \hat{i} - 7 \hat{j} , 16 \hat{i} - 19 \hat{j} - 4 \hat{k} , 3 \hat{j} - 6 \hat{k} , 2 \hat{i} - 5 \hat{j} + 10 \hat{k}$ are coplanar.

Q 5.1 | Page 65

Prove that the following vectors are coplanar:
$2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}$

Q 5.2 | Page 65

Prove that the following vectors are coplanar:
$\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}$

Q 6.1 | Page 65

Prove that the following vectors are non-coplanar:

$3 \hat{i} + \hat{j} - \hat{k} , 2 \hat{i} - \hat{j} + 7 \hat{k}\text{ and }7 \hat{i} - \hat{j} + 23 \hat{k}$
Q 6.2 | Page 65

Prove that the following vectors are non-coplanar:

$\hat{i} + 2 \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\hat{i} + \hat{j} + \hat{k}$
Q 7.1 | Page 66

If $\vec{a}$, $\vec{a}$, $\vec{c}$ are non-coplanar vectors, prove that the following vectors are non-coplanar: $2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}$

Q 7.2 | Page 66

If $\vec{a}$, $\vec{a}$, $\vec{c}$ are non-coplanar vectors, prove that the following vectors are non-coplanar: $\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}$

Q 8 | Page 66

Show that the vectors $\vec{a,} \vec{b,} \vec{c}$ given by $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\vec{c} = \hat{i} + \hat{j} + \hat{k}$  are non-coplanar.
Express vector $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\vec{c} = \hat{i} + \hat{j} + \hat{k}$  as a linear combination of the vectors $\vec{a,} \vec{b}\text{ and }\vec{c} .$

Q 9 | Page 66

Prove that a necessary and sufficient condition for three vectors $\vec{a}$, $\vec{b}$, $\vec{c}$  to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that $l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .$

Q 10 | Page 66

Show that the four points A, B, C and D with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$, $\vec{d}$ respectively are coplanar if and only if $3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .$

#### Pages 73 - 74

Q 1 | Page 73

Can a vector have direction angles 45°, 60°, 120°?

Q 2 | Page 73

Prove that 1, 1, 1 cannot be direction cosines of a straight line.

Q 3 | Page 73

A vector makes an angle of $\frac{\pi}{4}$ with each of x-axis and y-axis. Find the angle made by it with the z-axis.

Q 4 | Page 73

A vector $\vec{r}$ is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |$\vec{r}$| = 6 units, find $\vec{r}$.

Q 5 | Page 73

A vector $\vec{r}$ is inclined to -axis at 45° and y-axis at 60°.
If $\vec{r}$ = 8 units, find $\vec{r}$.

Q 6.1 | Page 73

Find the direction cosines of the following vectors:
$2 \hat{i} + 2 \hat{j} - \hat{k}$

Q 6.2 | Page 73

Find the direction cosines of the following vectors:
$6 \hat{i} - 2 \hat{j} - 3 \hat{k}$

Q 6.3 | Page 73

Find the direction cosines of the following vectors:
$3 \hat{i} - 4 \hat{k}$

Q 7.1 | Page 73

Find the angles at which the following vectors are inclined to each of the coordinate axes:
$\hat{i} - \hat{j} + \hat{k}$

Q 7.2 | Page 73

Find the angles at which the following vectors are inclined to each of the coordinate axes:
$\hat{j} - \hat{k}$

Q 7.3 | Page 73

Find the angles at which the following vectors are inclined to each of the coordinate axes:
$4 \hat{i} + 8 \hat{j} + \hat{k}$

Q 8 | Page 73

Show that the vector $\hat{i} + \hat{j} + \hat{k}$ is equally inclined with the axes OX, OY and OZ.

Q 9 | Page 73

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .$

Q 10 | Page 73

If a unit vector $\vec{a}$ makes an angle $\frac{\pi}{3}$ with $\hat{i} , \frac{\pi}{4}$ with $\hat{j}$  and an acute angle θ with $\hat{k}$, then find θ and hence, the components of $\vec{a}$.

Q 11 | Page 74

Find a vector $\vec{r}$ of magnitude $3\sqrt{2}$ units which makes an angle of $\frac{\pi}{4}$ and $\frac{\pi}{4}$ with y and z-axes respectively.

Q 12 | Page 74

A vector $\vec{r}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{r}$ is $2\sqrt{3}$, find $\vec{r}$.

#### Pages 75 - 77

Q 1 | Page 75

Define "zero vector".

Q 2 | Page 75

Define unit vector.

Q 3 | Page 75

Define position vector of a point.

Q 4 | Page 75

Write $\vec{PQ} + \vec{RP} + \vec{QR}$ in the simplified form.

Q 5 | Page 75

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors such that $x \vec{a} + y \vec{b} = \vec{0} ,$ then write the values of x and y.

Q 6 | Page 75

If $\vec{a}$ and $\vec{b}$ represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.

Q 7 | Page 75

If $\vec{a}$, $\vec{b}$, $\vec{c}$ represent the sides of a triangle taken in order, then write the value of $\vec{a} + \vec{b} + \vec{c} .$

Q 8 | Page 75

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are position vectors of the vertices A, B and C respectively, of a triangle ABC, write the value of $\vec{AB} + \vec{BC} + \vec{CA} .$

Q 9 | Page 75

If $\vec{a}$, $\vec{b}$, $\vec{c}$  are position vectors of the points A, B and C respectively, write the value of $\vec{AB} + \vec{BC} + \vec{AC} .$

Q 10 | Page 75

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are the position vectors of the vertices of a triangle, then write the position vector of its centroid.

Q 11 | Page 75

If G denotes the centroid of ∆ABC, then write the value of $\vec{GA} + \vec{GB} + \vec{GC} .$

Q 12 | Page 75

If $\vec{a}$ and $\vec{b}$ denote the position vectors of points A and B respectively and C is a point on ABsuch that 3AC = 2AB, then write the position vector of C.

Q 13 | Page 75

If D is the mid-point of side BC of a triangle ABC such that $\vec{AB} + \vec{AC} = \lambda \vec{AD} ,$ write the value of λ.

Q 14 | Page 75

If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of $\vec{AD} + \vec{BE} + \vec{CF} .$

Q 15 | Page 75

If $\vec{a}$ is a non-zero vector of modulus a and m is a non-zero scalar such that m $\vec{a}$ is a unit vector, write the value of m.

Q 16 | Page 75

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then write the value of $\vec{a} + \vec{b} + \vec{c} .$

Q 17 | Page 75

Write a unit vector making equal acute angles with the coordinates axes.

Q 18 | Page 75

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin2 α + sin2 β + sin2 γ.

Q 19 | Page 75

Write a vector of magnitude 12 units which makes 45° angle with X-axis, 60° angle with Y-axis and an obtuse angle with Z-axis.

Q 20 | Page 76

Write the length (magnitude) of a vector whose projections on the coordinate axes are 12, 3 and 4 units.

Q 21 | Page 76

Write the position vector of a point dividing the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ externally in the ratio 1 : 4, where $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\vec{b} = - \hat{i} + \hat{j} + \hat{k} .$

Q 22 | Page 76

Write the direction cosines of the vector $\vec{r} = 6 \hat{i} - 2 \hat{j} + 3 \hat{k} .$

Q 23 | Page 76

If $\vec{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} \text{ and }\vec{c} = \hat{k} + \hat{i} ,$ write unit vectors parallel to $\vec{a} + \vec{b} - 2 \vec{c} .$

Q 24 | Page 76

If $\vec{a} = \hat{i} + 2 \hat{j} , \vec{b} = \hat{j} + 2 \hat{k} ,$ write a unit vector along the vector $3 \vec{a} - 2 \vec{b} .$

Q 25 | Page 76

Write the position vector of a point dividing the line segment joining points having position vectors $\hat{i} + \hat{j} - 2 \hat{k} \text{ and }2 \hat{i} - \hat{j} + 3 \hat{k}$ externally in the ratio 2:3.

Q 26 | Page 76

If $\vec{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} , \vec{c} = \hat{k} + \hat{i}$, find the unit vector in the direction of $\vec{a} + \vec{b} + \vec{c}$.

Q 27 | Page 76
$\text{ If } \vec{a} = 3 \hat{i} - \hat{j} - 4 \hat{k} , \vec{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \text{ and }\vec{c} = \hat{i} + 2 \hat{j} - \hat{k} ,\text{ find }\left| 3 \vec{a} - 2 \vec{b} + 4 \vec{c} \right| .$
Q 28 | Page 76

A unit vector $\vec{r}$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{2}$ with $\hat{j}\text{ and }\hat{k}$  respectively and an acute angle θ with $\hat{i}$. Find θ.

Q 29 | Page 76

Write a unit vector in the direction of $\vec{a} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} .$

Q 30 | Page 76

If $\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} \text{ and }\vec{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} ,$  find a unit vector parallel to $\vec{a} + \vec{b}$.

Q 31 | Page 76

Write a unit vector in the direction of $\vec{b} = 2 \hat{i} + \hat{j} + 2 \hat{k}$.

Q 32 | Page 76

Find the position vector of the mid-point of the line segment AB, where A is the point (3, 4, −2) and B is the point (1, 2, 4).

Q 33 | Page 76

Find a vector in the direction of $\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} ,$ which has magnitude of 6 units.

Q 34 | Page 76

What is the cosine of the angle which the vector $\sqrt{2} \hat{i} + \hat{j} + \hat{k}$ makes with y-axis?

Q 35 | Page 76

Write two different vectors having same magnitude.

Q 36 | Page 76

Write two different vectors having same direction.

Q 37 | Page 76

Write a vector in the direction of vector $5 \hat{i} - \hat{j} + 2 \hat{k}$ which has magnitude of 8 unit.

Q 38 | Page 76

Write the direction cosines of the vector $\hat{i} + 2 \hat{j} + 3 \hat{k}$.

Q 39 | Page 76

Find a unit vector in the direction of $\vec{a} = 2 \hat{i} - 3 \hat{j} + 6 \hat{k}$.

Q 40 | Page 76

For what value of 'a' the vectors $2 \hat{i} - 3 \hat{j} + 4 \hat{k} \text{ and }a \hat{i} + 6 \hat{j} - 8 \hat{k}$  are collinear?

Q 41 | Page 76

Write the direction cosines of the vectors $- 2 \hat{i} + \hat{j} - 5 \hat{k}$.

Q 42 | Page 76

Find the sum of the following vectors: $\vec{a} = \hat{i} - 2 \hat{j} , \vec{b} = 2 \hat{i} - 3 \hat{j} , \vec{c} = 2 \hat{i} + 3 \hat{k} .$

Q 43 | Page 76

Find a unit vector in the direction of the vector $\vec{a} = 3 \hat{i} - 2 \hat{j} + 6 \hat{k}$.

Q 44 | Page 77

If $\vec{a} = x \hat{i} + 2 \hat{j} - z \hat{k}\text{ and }\vec{b} = 3 \hat{i} - y \hat{j} + \hat{k}$  are two equal vectors, then write the value of x + y + z.

Q 45 | Page 77

Write a unit vector in the direction of the sum of the vectors $\vec{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}$.

Q 46 | Page 77

Find the value of 'p' for which the vectors $3 \hat{i} + 2 \hat{j} + 9 \hat{k}$ and $\hat{i} - 2p \hat{j} + 3 \hat{k}$ are parallel.

Q 47 | Page 77

Find a vector $\vec{a}$ of magnitude $5\sqrt{2}$, making an angle of $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{2}$ with y-axis and an acute angle θ with z-axis.

Q 48 | Page 77

Write a unit vector in the direction of $\vec{PQ}$, where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.

Q 49 | Page 77

Find a vector in the direction of vector $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ which has magnitude 21 units.

Q 50 | Page 77

If $\left| \vec{a} \right| = 4$ and $- 3 \leq \lambda \leq 2$, then write the range of $\left| \lambda \vec{a} \right|$.

Q 51 | Page 77

In a triangle OAC, if B is the mid-point of side AC and $\vec{OA} = \vec{a} , \vec{OB} = \vec{b}$, then what is $\vec{OC}$.

Q 52 | Page 77

Write the position vector of the point which divides the join of points with position vectors $3 \vec{a} - 2 \vec{b}\text{ and }2 \vec{a} + 3 \vec{b}$ in the ratio 2 : 1.

#### Pages 78 - 79

Q 1 | Page 78

If in a ∆ABC, A = (0, 0), B = (3, 3 $\sqrt{3}$), C = (−3$\sqrt{3}$, 3), then the vector of magnitude 2 $\sqrt{2}$ units directed along AO, where O is the circumcentre of ∆ABC is

$\left( 1 - \sqrt{3} \right) \hat{i} + \left( 1 + \sqrt{3} \right) \hat{j}$

$\left( 1 + \sqrt{3} \right) \hat{i} + \left( 1 - \sqrt{3} \right) \hat{j}$

$\left( 1 + \sqrt{3} \right) \hat{i} + \left( \sqrt{3} - 1 \right) \hat{j}$

none of these

Q 2 | Page 78

If $\vec{a} , \vec{b}$ are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is

$\vec{a} + \vec{b}$

$\vec{a} - \vec{b}$

$\vec{b} - \vec{a}$

$- \left( \vec{a} + \vec{b} \right)$

Q 3 | Page 78

Forces 3 O $\vec{A}$, 5 O $\vec{B}$ act along OA and OB. If their resultant passes through C on AB, then

C is a mid-point of AB

C divides AB in the ratio 2 : 1

3 AC = 5 CB

2 AC = 3 CB

Q 4 | Page 78

If $\vec{a} , \vec{b} , \vec{c}$ are three non-zero vectors, no two of which are collinear and the vector $\vec{a} + \vec{b}$ is collinear with $\vec{c} , \vec{b} + \vec{c}$ is collinear with $\vec{a} ,$ then $\vec{a} + \vec{b} + \vec{c} =$

$\vec{a}$

$\vec{b}$

$\vec{c}$

none of these

Q 5 | Page 78

If points A (60 $\hat{i}$ + 3 $\hat{j}$), B (40 $\hat{i}$ − 8 $\hat{j}$) and C (a $\hat{i}$ − 52 $\hat{j}$) are collinear, then a is equal to

40

−40

20

−20

Q 6 | Page 78

If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then $O \vec{A} + O \vec{B} + O \vec{C} + O \vec{D} =$

$2 \vec{OG}$

$4 \vec{OG}$

$5 \vec{OG}$

$3 \vec{OG}$
Q 7 | Page 78

The vector cos α cos β $\hat{i}$ + cos α sin β $\hat{j}$ + sin α $\hat{k}$ is a

null vector

unit vector

constant vector

none of these

Q 8 | Page 78

In a regular hexagon ABCDEF, A $\vec{B}$ = a, B $\vec{C}$ = $\vec{b}\text{ and }\vec{CD} = \vec{c}$.
Then, $\vec{AE}$ =

$\vec{a} + \vec{b} + \vec{c}$

$2 \vec{a} + \vec{b} + \vec{c}$

$\vec{b} + \vec{c}$

$\vec{a} + 2 \vec{b} + 2 \vec{c}$

Q 9 | Page 78

The vector equation of the plane passing through $\vec{a} , \vec{b} , \vec{c} ,\text{ is }\vec{r} = \alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} ,$ provided that

α + β + γ = 0

α + β + γ =1

α + β = γ

α2 + β2 + γ2 = 1

Q 10 | Page 78

If O and O' are circumcentre and orthocentre of ∆ ABC, then $\vec{OA} + \vec{OB} + \vec{OC}$ equals

2$\vec{OO}$

$O \vec{O'}$
$O \vec{O'}$

$2 \vec{O'O}$
Q 11 | Page 78

If $\vec{a}$, $\vec{a}$, $\vec{c}$ and $\vec{d}$ are the position vectors of points A, B, C, D such that no three of them are collinear and $\vec{a} + \vec{c} = \vec{b} + \vec{d} ,$ then ABCD is a

rhombus

rectangle

square

parallelogram

Q 12 | Page 79

Let G be the centroid of ∆ ABC. If $\vec{AB} = \vec{a,} \vec{AC} = \vec{b,}$ then the bisector $\vec{AG} ,$ in terms of $\vec{a}\text{ and }\vec{b}$ is

$\frac{2}{3}\left( \vec{a} + \vec{b} \right)$

$\frac{1}{6}\left( \vec{a} + \vec{b} \right)$
$\frac{1}{3}\left( \vec{a} + \vec{b} \right)$

$\frac{1}{2}\left( \vec{a} + \vec{b} \right)$
Q 13 | Page 79

If ABCDEF is a regular hexagon, then $\vec{AD} + \vec{EB} + \vec{FC}$ equals

$2 \vec{AB}$

$\vec{0}$

$3 \vec{AB}$

$4 \vec{AB}$
Q 14 | Page 79

The position vectors of the points ABC are $2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k}\text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}$ respectively.
These points

form an isosceles triangle

form a right triangle

are collinear

form a scalene triangle

Q 15 | Page 79

If three points A, B and C have position vectors $\hat{i} + x \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k}\text{ and }y \hat{i} - 2 \hat{j} - 5 \hat{k}$ respectively are collinear, then (x, y) =

(2, −3)

(−2, 3)

(−2, −3)

(2, 3)

Q 16 | Page 79

ABCD is a parallelogram with AC and BD as diagonals.
Then, $\vec{AC} - \vec{BD} =$

$4 \vec{AB}$

$3 \vec{AB}$
$2 \vec{AB}$

$\vec{AB}$
Q 17 | Page 79

If OACB is a parallelogram with $\vec{OC} = \vec{a}\text{ and }\vec{AB} = \vec{b} ,$ then $\vec{OA} =$

$\left( \vec{a} + \vec{b} \right)$

$\left( \vec{a} - \vec{b} \right)$

$\frac{1}{2}\left( \vec{b} - \vec{a} \right)$

$\frac{1}{2}\left( \vec{a} - \vec{b} \right)$

Q 18 | Page 79

If $\vec{a}\text{ and }\vec{b}$ are two collinear vectors, then which of the following are incorrect?

$\vec{b} = \lambda \vec{a}$ for some scalar λ

$\vec{a} = \pm \vec{b}$

the respective components of $\vec{a}\text{ and }\vec{b}$ are proportional

both the vectors $\vec{a}\text{ and }\vec{b}$ have the same direction but different magnitudes

#### Pages 29 - 33

Q 1.1 | Page 29

Find $\vec{a} \cdot \vec{b}$ when

$\vec{a} =\hat{i} - 2\hat{j} + \hat{k}\text{ and } \vec{b} = 4 \hat{i} - 4\hat{j} + 7 \hat{k}$

Q 1.2 | Page 29

Find $\vec{a} \cdot \vec{b}$ when

$\vec{a} = \hat{j} + 2 \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{k}$

Q 1.3 | Page 29

Find $\vec{a} \cdot \vec{b}$ when

$\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}$

Q 2.1 | Page 30

For what value of λ are the vectors $\vec{a} \text{ and }\vec{b}$ perpendicular to each other if $\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 4\hat{i} - 9 \hat{j} + 2\hat{k}$

Q 2.2 | Page 30

For what value of λ are the vectors $\vec{a} and \vec{b}$ perpendicular to each other if

$\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}$

Q 2.3 | Page 30

For what value of λ are the vectors $\vec{a} \text{ and } \vec{b}$ perpendicular to each other if $\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 5 \hat{i} - 9 \hat{j} + 2\hat{k}$

Q 2.4 | Page 30

For what value of λ are the vectors $\vec{a} \text{ and } \vec{b}$ perpendicular to each other if

$\vec{a} = \lambda \hat{i} + 3 \hat{j} + 2 \hat{k}\text { and } \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}$

Q 3 | Page 30

If $\vec{a} \text{ and } \vec{b}$ are two vectors such that $\left| \vec{a} \right| = 4, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 6$ find the angle between $\vec{a} \text{ and } \vec{b} .$

Q 4 | Page 30

$\text{ If } \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + 2\hat{k} , \text{find} \left( \vec{a} - 2 \vec{b} \right) \cdot \left( \vec{a} + \vec{b} \right) .$

Q 5.1 | Page 30

Find the angle between the vectors $\vec{a} \text{ and } \vec{b}$ where $\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = \hat{j} + \hat{k}$

Q 5.2 | Page 30

Find the angle between the vectors $\vec{a} \text{ and } \vec{b}$ $\vec{a} = 3\hat{i} - 2\hat{j} - 6\hat{k} \text{ and } \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}$

Q 5.3 | Page 30

Find the angle between the vectors $\vec{a} \text{ and } \vec{b}$  $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}$

Q 5.4 | Page 30

Find the angle between the vectors $\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$

Q 5.5 | Page 30

Find the angle between the vectors $\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}$

Q 6 | Page 30

Find the angles which the vector $\vec{a} = \hat{i} -\hat {j} + \sqrt{2} \hat{k}$ makes with the coordinate axes.

Q 7.1 | Page 30

Dot product of a vector with $\hat{i} + \hat{j} - 3\hat{k} , \hat{i} + 3\hat{j} - 2 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 4 \hat{k}$ are 0, 5 and 8 respectively. Find the vector.

Q 7.2 | Page 30

Dot products of a vector with vectors $\hat{i} - \hat{j} + \hat{k} , 2\hat{ i} + \hat{j} - 3\hat{k} \text{ and } \text{i} + \hat{j} + \hat{k}$  are respectively 4, 0 and 2. Find the vector.

Q 8.1 | Page 30

If  $\hat{a} \text{ and } \hat{b}$ are unit vectors inclined at an angle θ, prove that $\cos\frac{\theta}{2} = \frac{1}{2}\left| \hat{a} + \hat{b} \right|$

Q 8.2 | Page 30

If $\hat{a} \text{ and } \hat{b}$ $\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}$

Q 9 | Page 30

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $\sqrt{3}$.

Q 10 | Page 30

If $\vec{a,} \vec{b,} \vec{c}$ are three mutually perpendicular unit vectors, then prove that $\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}$

Q 11 | Page 30

If $\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|$

Q 12 | Page 30

Show that the vector $\hat{i} + \hat{j} + \hat{k}$ is equally inclined to the coordinate axes.

Q 13 | Page 30

Show that the vectors $\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3 hat{i} - 6 hat{j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 \hat{k} \right)$ mutually perpendicular unit vectors.

Q 14 | Page 30

For any two vectors $\vec{a} \text{ and } \vec{b}$ show that $\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 0 \Leftrightarrow \left| \vec{a} \right| = \left| \vec{b} \right|$

Q 15 | Page 30

If $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$  $\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}$  $\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}$ find λ such that $\vec{a}$ is perpendicular to $\lambda \vec{b} + \vec{c}$

Q 16 | Page 30

If $\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,$ then find the value of λ, so that $\vec{p} + \vec{q}$ and $\vec{p} - \vec{q}$  are perpendicular vectors.

Q 17 | Page 30

If $\vec{\alpha} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \text{ and } \vec{\beta} = 2 \hat{i} + \hat{j} - 4 \hat{k} ,$ then express $\vec{\beta}$ in the form of  $\vec{\beta} = \vec{\beta_1} + \vec{\beta_2} ,$  where $\vec{\beta_1}$ is parallel to $\vec{\alpha} \text{ and } \vec{\beta_2}$  is perpendicular to $\vec{\alpha}$

Q 18 | Page 31

If either $\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}$  then $\vec{a} \cdot \vec{b} = 0 .$ But the converse need not be true. Justify your answer with an example.

Q 19 | Page 31

Show that the vectors $\vec{a} = 3 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} - 3 \hat{j} + 5 \hat{k} , \vec{c} = 2 \hat{i} + \hat{j} - 4 \hat{k}$ form a right-angled triangle.

Q 20 | Page 31

If $\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}$ $\vec{a} + \lambda \vec{b}$ is perpendicular to $\vec{c}$ then find the value of λ.

Q 21 | Page 31

Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1).

Q 22 | Page 31

Find the magnitude of two vectors $\vec{a} \text{ and } \vec{b}$ that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.

Q 23 | Page 31

Show that the points whose position vectors are $\vec{a} = 4 \hat{i} - 3 \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - 4 \hat{j} + 5 \hat{k} , \vec{c} = \hat{i} - \hat{j}$ form a right triangle.

Q 24 | Page 31

If the vertices Aand C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC

Q 25 | Page 31

If AB and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C

Q 26 | Page 31

Find the projection of $\vec{b} + \vec{c} \text { on }\vec{a}$  where $\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .$

Q 27 | Page 31

If $\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 hat{k} ,$ then show that the vectors $\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}$ are orthogonal.

Q 28 | Page 31

A unit vector $\vec{a}$ makes angles $\frac{\pi}{4}and\frac{\pi}{3}$ with $\hat{i}$ and$\hat{j}$  respectively and an acute angle θ with $\hat{k}$ Find the angle θ and components of $\vec{a}$

Q 29 | Page 31

If two vectors $\vec{a} \text{ and } \vec{b}$ are such that $\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 1 \text{ and } \vec{a} \cdot \vec{b} = 1,$  then find the value of $\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .$

Q 30.1 | Page 31

If $\vec{a}$ is a unit vector, then find $\left| \vec{x} \right|$  in each of the following.

$\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8$

Q 30.2 | Page 31

If $\vec{a}$ is a unit vector, then find $\left| \vec{x} \right|$  in each of the following.

$\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12$

Q 31.1 | Page 31

Find $\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|$ if

$\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|$

Q 31.2 | Page 31

Find  $\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|$ if

$\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|$

Q 31.3 | Page 31

Find $\left| \vec{a} \right| and \left| \vec{b} \right|$ if

$\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3 and \left| \vec{a} \right| = 2\left| \vec{b} \right|$

Q 32.1 | Page 31

Find $\left| \vec{a} - \vec{b} \right|$ if

$\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} \cdot \vec{b} = 8$

Q 32.2 | Page 31

Find $\left| \vec{a} - \vec{b} \right|$

$\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 4 \text{ and } \vec{a} \cdot \vec{b} = 1$

Q 32.3 | Page 31

Find $\left| \vec{a} - \vec{b} \right|$ if

$\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 4$

Q 33.1 | Page 31

Find the angle between two vectors $\vec{a} \text{ and } \vec{b}$ if

$\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = 2 \text{ and } \vec{a} \cdot \vec{b} = \sqrt{6}$

Q 33.2 | Page 31

Find the angle between two vectors $\vec{a} \text{ and } \vec{b}$

$\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 1$

Q 34 | Page 32

Express the vector $\vec{a} = 5 \text{i} - 2 \text{j} + 5 \text{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b} = 3 \text{i} + \text{k}$  and other is perpendicular to $\vec{b}$

Q 35 | Page 32

If $\vec{a} \text{ and } \vec{b}$ are two vectors of the same magnitude inclined at an angle of 30°, such that $\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .$

Q 36 | Page 32

Express $2 \hat{i} - \hat{j} + 3 \hat{k}$ as the sum of a vector parallel and a vector perpendicular to $2 \hat{i} + 4 \hat{j} - 2 \hat{k} .$

Q 37 | Page 32

Decompose the vector $6 \hat{i} - 3 \hat{j} - 6 \hat{k}$ into vectors which are parallel and perpendicular to the vector $\hat{i} + \hat{j} + \hat{k} .$

Q 38 | Page 32

Let $\vec{a} = 5 \hat{i} - \hat{j} + 7 \hat{k} \text{ and } \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} .$ Find λ such that $\vec{a} + \vec{b}$ is orthogonal to $\vec{a} - \vec{b}$

Q 39 | Page 32

If $\vec{a} \cdot \vec{a} = 0 \text{ and } \vec{a} \cdot \vec{b} = 0,$ what can you conclude about the vector $\vec{b}$ ?

Q 40 | Page 32

If $\vec{c}$ s perpendicular to both $\vec{a} \text{ and } \vec{b}$ then prove that it is perpendicular to both $\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}$

Q 41 | Page 32

If $\left| \vec{a} \right| = a \text{ and } \left| \vec{b} \right| = b,$ prove that $\left( \frac{\vec{a}}{a^2} - \frac{\vec{b}}{b^2} \right)^2 = \left( \frac{\vec{a} - \vec{b}}{ab} \right)^2 .$

Q 42 | Page 32

If $\vec{a,} \vec{b,} \vec{c}$  are three non-coplanar vectors, such that $\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c} = 0,$ then show that $\vec{d}$ is the null vector.

Q 43 | Page 32

If a vector $\vec{a}$ is perpendicular to two non-collinear vectors $\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}$ is perpendicular to every vector in the plane of $\vec{b} \text{ and } \vec{c} .$

Q 44 | Page 32

If $\vec{a} + \vec{b} + \vec{c} = \vec{0} ,$ show that the angle θ between the vectors $\vec{b} \text{ and } \vec{c}$ is given by  $\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .$

Q 45 | Page 32

Let $\vec{u,} \vec{v} \text{ and } \vec{w}$  be vectors such $\vec{u} + \vec{v} + \vec{w} = \vec{0} .$ If $\left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5,$ then find $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .$

Q 46 | Page 32

Let $\vec{a} = x^2 \hat{i} + 2 \hat{j} - 2 \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{c} = x^2 \hat{i} + 5 \hat{j} - 4 \hat{k}$ be three vectors. Find the values of x for which the angle between $\vec{a} \text{ and } \vec{b}\ \text{ is acute and the angle between \[\vec{b} \text{ and } \vec{c}$ is obtuse.

Q 48 | Page 32

Find the values of x and y if the vectors $\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}$ are mutually perpendicular vectors of equal magnitude.

Q 49 | Page 33

If $\vec{a}$ $\vec{b}$  are two vectors such that $\left| \vec{a} + \vec{b} \right| = \left| \vec{b} \right|$ then prove that $\vec{a} + 2 \vec{b}$ is perpendicular to $\vec{a}$

#### Page 46

Q 1 | Page 46

In a triangle OAB,$\angle$AOB = 90º. If P and Q are points of trisection of AB, prove that ${OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2$

Q 2 | Page 46

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Q 3 | Page 46

(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Q 4 | Page 46

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Q 5 | Page 46

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

Q 6 | Page 46

Prove that the diagonals of a rhombus are perpendicular bisectors of each other.

Q 7 | Page 46

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

Q 8 | Page 46

If AD is the median of ∆ABC, using vectors, prove that ${AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)$

Q 9 | Page 46

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.

Q 10 | Page 46

In a quadrilateral ABCD, prove that ${AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 = {AC}^2 + {BD}^2 + 4 {PQ}^2$ where P and Q are middle points of diagonals AC and BD.

#### Pages 46 - 49

Q 1 | Page 46

What is the angle between vectors $\vec{a} \text{ and } \vec{b}$ with magnitudes 2 and $\sqrt{3}$ respectively? Given $\vec{a} . \vec{b} = \sqrt{3} .$

Q 2 | Page 46

$\vec{a} \text{ and } \vec{b}$ are two vectors such that $\vec{a} . \vec{b} = 6, \left| \vec{a} \right| = 3 \text{ and } \left| \vec{b} \right| = 4 .$ Write the projection of $\vec{a} \text{ on } \vec{b}$

Q 3 | Page 49

If $\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,$ then the angle between $\vec{a} \text{ and } \vec{b}$ is

(a) $\frac{\pi}{6}$

(b) $\frac{2\pi}{3}$

(c) $\frac{5\pi}{3}$

(d) $\frac{\pi}{3}$

Q 3 | Page 46

Find the cosine of the angle between the vectors $4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k}] .$

Q 4 | Page 46

If the vectors $3 \hat{i} + m \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} - 8 \hat{k}$  are orthogonal, find m

Q 5 | Page 46

If the vectors $3 \hat{i} - 2 \hat{j} - 4 \hat{k}\text{ and } 18 \hat{i} - 12 \hat{j}] - m \hat{k}$ are parallel, find the value of m.

Q 6 | Page 46

If $\vec{a} \text{ and } \vec{b}$ are vectors of equal magnitude, write the value of $\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) .$

Q 7 | Page 47

If $\vec{a} \text{ and } \vec{b}$ are two vectors such that $\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0,$ find the relation between the magnitudes of $\vec{a} \text{ and } \vec{b}$

Q 8 | Page 47

For any two vectors $\vec{a} \text{ and } \vec{b}$ write when $\left| \vec{a} + \vec{b} \right| = \left| \vec{a} \right| + \left| \vec{b} \right|$ holds.

Q 9 | Page 47

For any two vectors $\vec{a} \text{ and } \vec{b}$ write when $\left| \vec{a} + \vec{b} \right| = \left| \vec{a} - \vec{b} \right|$ holds.

Q 10 | Page 47

If $\vec{a} \text{ and } \vec{b}$ are two vectors of the same magnitude inclined at an angle of 60° such that $\vec{a} . \vec{b} = 8,$ write the value of their magnitude.

Q 11 | Page 47

If $\vec{a} . \vec{a} = 0 \text{ and } \vec{a} . \vec{b} = 0,$ what can you conclude about the vector $\vec{b}$

Q 12 | Page 47

If $\vec{b}$ is a unit vector such that$\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8, \text{ find } \left| \vec{a} \right| .$

Q 13 | Page 47

If $\hat{a} , \hat{b}$ are unit vectors such that $\hat{a} + \hat{b}$  is a unit vector, write the value of $\left| \hat{a} - \hat{b} \right| .$

Q 14 | Page 47

If $\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .$

Q 15 | Page 47

If $\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,$  find the projection of $\vec{a} \text{ on } \vec{b}$

Q 16 | Page 47

For any two non-zero vectors, write the value of $\frac{\left| \vec{a} + \vec{b} \right|^2 + \left| \vec{a} - \vec{b} \right|^2}{\left| \vec{a} \right|^2 + \left| \vec{b} \right|^2} .$

Q 17 | Page 47

Write the projections of $\vec{r} = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$ on the coordinate axes.

Q 18 | Page 47

Write the component of $\vec{b}$ along $\vec{a}$

Q 19 | Page 47

Write the value of $\left( \vec{a} . \hat{i} \right) \hat{i} + \left( \vec{a} . \hat{j} \right) \hat{j} + \left( \vec{a} . \hat{k} \right) \hat{k} ,$  where $\vec{a}$ is any vector.

Q 20 | Page 47

Find the value of θ ∈(0, π/2) for which vectors $\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}$ are perpendicular.

Q 21 | Page 47

Write the projection of $\hat{i} + \hat{j} + \hat{k}$ along the vector $\hat{j}$

Q 22 | Page 47

Write a vector satisfying $\vec{a} . \hat{i} = \vec{a} . \left( \hat{i} + \hat{j} \right) = \vec{a} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 1 .$

Q 23 | Page 47

If $\vec{a} \text{ and } \vec{b}$ are unit vectors, find the angle between $\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} .$

Q 24 | Page 47

If $\vec{a} \text{ and } \vec{b}$ are mutually perpendicular unit vectors, write the value of $\left| \vec{a} + \vec{b} \right| .$

Q 25 | Page 47

If $\vec{a} , \vec{b} \text{ and } \vec{c}$ are mutually perpendicular unit vectors, write the value of $\left| \vec{a} + \vec{b} + \vec{c} \right| .$

Q 26 | Page 47

Find the angle between the vectors $\vec{a} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - \hat{k} .$

Q 27 | Page 47

For what value of λ are the vectors $\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ perpendicular to each other?

Q 28 | Page 47

Find the projection of $\vec{a} \text{ on } \vec{b} \text{ if } \vec{a} \cdot \vec{b} = 8 \text{ and } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} .$

Q 29 | Page 47

Write the value of p for which $\vec{a} = 3 \hat{i} + 2 \hat{j}] + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}$

Q 30 | Page 47

Find the value of λ if the vectors $2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 2 \hat{j} - 4 \hat{k}$ are perpendicular to each other.

Q 31 | Page 48

If $\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 3,$ find the projection of $\vec{b} \text{ on } \vec{a}$

Q 32 | Page 48

Write the angle between two vectors $\vec{a} \text{ and } \vec{b}$ with magnitudes $\sqrt{3}$ and 2 respectively if $\vec{a} \cdot \vec{b} = \sqrt{6} .$

Q 33 | Page 48

Write the projection of the vector $\hat{i} + 3 \hat{j} + 7 \hat{k}$ on the vector $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$

Q 34 | Page 48

Find λ when the projection of $\vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \text{ on } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k}$  is 4 units.

Q 35 | Page 48

For what value of λ are the vectors $\vec{a} = 2 \text{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ perpendicular to each other?

Q 36 | Page 48

Write the projection of the vector $7 \hat{i} + \hat{j} - 4 \hat{k}$ on the vector $2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .$

Q 37 | Page 48

Write the value of λ so that the vectors $\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ are perpendicular to each other.

Q 38 | Page 48

Write the projection of $\vec{b} + \vec{c} \text{ on } \vec{a} \text{ when } \vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .$

Q 39 | Page 48

If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $\left| \vec{a} + \vec{b} \right| = 3$ and $\left| \vec{a} \right| = 5$ find the value of $\left| \vec{b} \right|$

Q 40 | Page 48

If the vectors $\vec{a}$  and $\vec{b}$ are such that $\left| \vec{a} \right| = 3, \left| \vec{b} \right| = \frac{2}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector, then write the angle between $\vec{a}$ and $\vec{b}$

Q 41 | Page 48

If $\vec{a}$ and $\vec{b}$ are two unit vectors such that $\vec{a} + \vec{b}$ is also a unit vector, then find the angle between $\vec{a}$ and $\vec{b}$

Q 42 | Page 48

If $\vec{a}$ and $\vec{b}$ are unit vectors, then find the angle between $\vec{a}$ and $\vec{b}$ given that $\left( \sqrt{3} \vec{a} - \vec{b} \right)$ is a unit vector.

#### Pages 49 - 51

Q 1 | Page 49

The vectors $\vec{a} \text{ and } \vec{b}$ satisfy the equations $2 \vec{a} + \vec{b} = \vec{p} \text{ and } \vec{a} + 2 \vec{b} = \vec{q} , \text{ where } \vec{p} = \hat{i} + \hat{j} \text{ and } \vec{q} = \hat{i} - \hat{j} .$ the angle between $\vec{a} \text{ and } \vec{b}$ then

(a) $\cos \theta = \frac{4}{5}$

(b) $\sin \theta = \frac{1}{\sqrt{2}}$

(c) $\cos \theta = - \frac{4}{5}$

(d) $\cos \theta = - \frac{3}{5}$

Q 2 | Page 49

If $\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,$  then $\vec{a} =$

(a) $\vec{0}$

(b) $\hat{i}$

(c)  $\hat{j}$

(d) $\hat{i} + \hat{j} + \hat{k}$

Q 3 | Page 49

If $\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,$ then the angle between $\vec{a} \text{ and } \vec{b}$ is

(a) $\frac{\pi}{6}$

(b) $\frac{2\pi}{3}$

(c) $\frac{5\pi}{3}$

(d) $\frac{\pi}{3}$

Q 4 | Page 49

Let $\vec{a} \text{ and } \vec{b}$  be two unit vectors and α be the angle between them. Then, $\vec{a} + \vec{b}$ is a unit vector if

(a) $\vec{a} + \vec{b}$

(b) $\alpha = \frac{\pi}{3}$

(c) $\alpha = \frac{2\pi}{3}$

(d) $\alpha = \frac{\pi}{2}$

Q 5 | Page 49

The vector (cos α cos β) $\hat{i}$ + (cos α sin β) $\hat{j}$ + (sin α) $\hat{k}$  is a

(a) null vector

(b) unit vector

(c) constant vector

(d) None of these

Q 6 | Page 49

If the position vectors of P and Q are $\hat{i} + 3 \hat{j} - 7 \hat{k} \text{ and } 5 \text{i} - 2 \hat{j} + 4 \hat{k}$ then the cosine of the angle between $\vec{PQ}$ and y-axis is

(a) $\frac{5}{\sqrt{162}}$

(b) $\frac{4}{\sqrt{162}}$

(c) $- \frac{5}{\sqrt{162}}$

(d) $\frac{11}{\sqrt{162}}$

Q 7 | Page 49

If $\vec{a} \text{ and } \vec{b}$ are unit vectors, then which of the following values of $\vec{a} . \vec{b}$ is not possible?

(a) $\sqrt{3}$

(b) $\sqrt{3}/2$

(c) $1/\sqrt{2}$

(d) −1/2

Q 8 | Page 49

If the vectors $\hat{i} - 2x \hat{j} + 3y \hat{k} \text{ and } \hat{i} + 2x \hat{j} - 3y \hat{k}$ are perpendicular, then the locus of (xy) is

(a) a circle

(b) an ellipse

(c) a hyperbola

(d) None of these

Q 9 | Page 49

The vector component of $\vec{b}$ perpendicular to $\vec{a}$ is

(a) $\left( \vec{b} . \vec{c} \right) \vec{a}$

(b) $\frac{\vec{a} \times \left( \vec{b} \times \vec{a} \right)}{\left| \vec{a} \right|^2}$

(c) $\vec{a} \times \left( \vec{b} \times \vec{a} \right)$

(d) None of these

Q 10 | Page 49

What is the length of the longer diagonal of the parallelogram constructed on $5 \vec{a} + 2 \vec{b} \text{ and } \vec{a} - 3 \vec{b}$ if it is given that $\left| \vec{a} \right| = 2\sqrt{2}, \left| \vec{b} \right| = 3$ and the angle between $\vec{a} \text{ and } \vec{b}$ is π/4?

(a) 15

(b) $\sqrt{113}$

(c) $\sqrt{593}$

(d) $\sqrt{369}$

Q 11 | Page 50

If $\vec{a}$ is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ $\vec{a}$ is a unit vector if

(a) λ = 1

(b) λ = −1

(c) a = |λ|

(d) $a = \frac{1}{\left| \lambda \right|}$

Q 12 | Page 49

If θ is the angle between two vectors $\vec{a} \text{ and } \vec{b} , \text{ then } \vec{a} \cdot \vec{b} \geq 0$  only when

(a) $0 < \theta < \frac{\pi}{2}$

(b) $0 \leq \theta \leq \frac{\pi}{2}$

(c) 0 < θ < π

(d) 0 ≤ θ ≤ π

Q 13 | Page 50

The values of x for which the angle between $\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$  is obtuse and the angle between $\vec{b}$ and the z-axis is acute and less than $\frac{\pi}{6}$

(a) $x > \frac{1}{2} or x < 0$

(b) $0 < x < \frac{1}{2}$

(c) $\frac{1}{2} < x < 15$

(d) ϕ

Q 14 | Page 50

If $\vec{a} , \vec{b} , \vec{c}$ are any three mutually perpendicular vectors of equal magnitude a, then $\left| \vec{a} + \vec{b} + \vec{c} \right|$ is equal to

(a)

(b) $\sqrt{2}a$

(c) $\sqrt{3}a$

(d) 2

(e) None of these

Q 15 | Page 50

If the vectors $3 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} + 8 \hat{k}$ are perpendicular, then λ is equal to

(a) −14

(b) 7

(c) 14

(d) $\frac{1}{7}$

Q 16 | Page 50

The projection of the vector $\hat{i} + \hat{j} + \hat{k}$ along the vector of $\hat{j}$

(a) 1

(b) 0

(c) 2

(d) −1

(e) −2

Q 17 | Page 50

The vectors $2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ and $a \hat{i} + \hat{b} j + c \hat{k}$ are perpendicular if

(a) a = 2, b = 3, c = −4

(b) a = 4, b = 4, c = 5

(c) a = 4, b = 4, c = −5

(d) a = −4, b = 4, c = −5

Q 18 | Page 50

If $\left| \vec{a} \right| = \left| \vec{b} \right|, \text{ then } \left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) =$

(a) positive

(b) negative

(c) 0

(d) None of these

Q 19 | Page 50

If $\vec{a} \text{ and } \vec{b}$ are unit vectors inclined at an angle θ, then the value of $\left| \vec{a} - \vec{b} \right|$

(a) $2 \sin\frac{\theta}{2}$

(b) 2 sin θ

(c) $2 \cos\frac{\theta}{2}$

(d) 2 cos θ

Q 20 | Page 50

If $\vec{a} and \vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}\left| \vec{a} + \vec{b} \right| + \left| \vec{a} - \vec{b} \right|$

(a) 2

(b) $2\sqrt{2}$

(c) 4

(d) None of these

Q 21 | Page 50

If the angle between the vectors $x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}$ is acute, then x lies in the interval

(a) (−4, 7)

(b) [−4, 7]

(c) R −[−4, 7]

(d) R −(4, 7)

Q 22 | Page 50

If $\vec{a} \text{ and } \vec{b}$ are two unit vectors inclined at an angle θ, such that $\left| \vec{a} + \vec{b} \right| < 1,$ then

(a) $\theta < \frac{\pi}{3}$

(b) $\theta > \frac{2\pi}{3}$

(c) $\frac{\pi}{3} < \theta < \frac{2\pi}{3}$

(d) $\frac{2\pi}{3} < \theta < \pi$

Q 23 | Page 50

Let $\vec{a} , \vec{b} , \vec{c}$ be three unit vectors, such that $\left| \vec{a} + \vec{b} + \vec{c} \right|$ =1 and $\vec{a}$ is perpendicular to $\vec{b}$  If $\vec{c}$ makes angles α and β with $\vec{a} and \vec{b}$ respectively, then cos α + cos β =

(a) $- \frac{3}{2}$

(b) $\frac{3}{2}$

(c) 1

(d) −1

Q 24 | Page 51

The orthogonal projection of $\vec{a} \text{ on } \vec{b}$ is

(a) $\frac{\left( \vec{a} \cdot \vec{b} \right) \vec{a}}{\left| \vec{a} \right|^2}$

(b) $\frac{\left( \vec{a} \cdot \vec{b} \right) \vec{b}}{\left| \vec{b} \right|^2}$

(c)  $\frac{\vec{a}}{\left| \vec{a} \right|}$

(d) $\frac{\vec{b}}{\left| \vec{b} \right|}$

Q 25 | Page 51

If θ is an acute angle and the vector (sin θ) $\text{i}$  + (cos θ) $\hat{j}$  is perpendicular to the vector $\hat{i} - \sqrt{3} \hat{j} ,$ then θ =

(a) $\frac{\pi}{6}$

(b) $\frac{\pi}{5}$

(c)  $\frac{\pi}{4}$

(d)  $\frac{\pi}{3}$

Q 26 | Page 51

If $\vec{a} \text{ and }\vec{b}$ be two unit vectors and θ the angle between them, then $\vec{a} + \vec{b}$ is a unit vector if θ =

(a) $\frac{\pi}{4}$

(b) $\frac{\pi}{3}$

(c) $\frac{\pi}{2}$

(d) $\frac{2\pi}{3}$

## R.D. Sharma solutions for Class 12 Mathematics chapter 24 - Scalar Or Dot Product

R.D. Sharma solutions for Class 12 Mathematics chapter 24 (Scalar Or Dot Product) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 24 Scalar Or Dot Product are Addition of Vectors, Multiplication of a Vector by a Scalar, Concept of Direction Cosines, Properties of Vector Addition, Geometrical Interpretation of Scalar, Scalar Triple Product of Vectors, Vector (Or Cross) Product of Two Vectors, Scalar (Or Dot) Product of Two Vectors, Position Vector of a Point Dividing a Line Segment in a Given Ratio, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Types of Vectors, Components of a Vector, Section formula, Vector Joining Two Points, Vectors Examples and Solutions, Projection of a Vector on a Line, Introduction of Product of Two Vectors.

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