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# RD Sharma solutions for Class 12 Mathematics chapter 23 - Algebra of Vectors

## Chapter 23: Algebra of Vectors

#### Chapter 23: Algebra of Vectors solutions [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

#### Chapter 23: Algebra of Vectors solutions [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.

#### Chapter 23: Algebra of Vectors solutions [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
$\frac{\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c}}{\alpha + \beta + \gamma},\text{ where }\alpha = \left| \vec{b} - \vec{c} \right|$

#### Chapter 23: Algebra of Vectors solutions [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.

#### Chapter 23: Algebra of Vectors solutions [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 $\lambda$

#### Chapter 23: Algebra of Vectors solutions [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.

#### Chapter 23: Algebra of Vectors solutions [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.

#### Chapter 23: Algebra of Vectors solutions [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 $3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .$

#### Chapter 23: Algebra of Vectors solutions [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 $\vec{r}$.

#### Chapter 23: Algebra of Vectors solutions [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.

#### Chapter 23: Algebra of Vectors solutions [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

## RD Sharma solutions for Class 12 Mathematics chapter 23 - Algebra of Vectors

RD Sharma solutions for Class 12 Maths chapter 23 (Algebra of Vectors) include all questions with solution and detail explanation. 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 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 23 Algebra of Vectors are Introduction of Product of Two Vectors, Projection of a Vector on a Line, 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.

Using RD Sharma Class 12 solutions Algebra of Vectors exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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