# RD Sharma solutions for Class 12 Maths chapter 23 - Algebra of Vectors [Latest edition]

#### Chapters ## Solutions for Chapter 23: Algebra of Vectors

Below listed, you can find solutions for Chapter 23 of CBSE, Karnataka Board PUC RD Sharma for Class 12 Maths.

Exercise 23.1Exercise 23.2Exercise 23.3Exercise 23.4Exercise 23.4Exercise 23.6Exercise 23.7Exercise 23.8Exercise 23.9Very Short AnswersMCQ
Exercise 23.1 [Page 4]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.1 [Page 4]

Exercise 23.1 | 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.

Exercise 23.1 | 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

Exercise 23.1 | 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

Exercise 23.1 | Q 4 | Page 4

In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.

Exercise 23.1 | Q 5.1 | Page 4

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

• True

• False

Exercise 23.1 | Q 5.2 | Page 4

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

• true

• False

Exercise 23.1 | Q 5.3 | Page 4

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

• true

• false

Exercise 23.1 | Q 5.4 | Page 4

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

• true

• false

Exercise 23.1 | Q 5.5 | Page 4

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

• true

• false

Exercise 23.2 [Page 17]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.2 [Page 17]

Exercise 23.2 | Q 1 | Page 17

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

Exercise 23.2 | Q 2 | Page 17

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

Exercise 23.2 | 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 $\vec{a}$ − $\vec{b}$.

Exercise 23.2 | 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 $\vec{a}$ ?

Exercise 23.2 | Q 5.1 | Page 17

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

Exercise 23.2 | Q 5.2 | Page 17

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

Exercise 23.2 | Q 5.3 | Page 17

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

Exercise 23.2 | Q 6 | Page 17

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

Exercise 23.2 | Q 7.1 | Page 17

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

Exercise 23.2 | Q 7.2 | Page 17

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

Exercise 23.2 | 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.

Exercise 23.2 | Q 9 | Page 17

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

Exercise 23.2 | Q 10 | Page 17

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

Exercise 23.3 [Pages 23 - 24]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.3 [Pages 23 - 24]

Exercise 23.3 | 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.

Exercise 23.3 | 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.

Exercise 23.3 | 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.

Exercise 23.3 | 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.

Exercise 23.3 | 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.

Exercise 23.3 | 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
$\frac{\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c}}{\alpha + \beta + \gamma},\text{ where }\alpha = \left| \vec{b} - \vec{c} \right|$

Exercise 23.4 [Pages 36 - 37]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.4 [Pages 36 - 37]

Exercise 23.4 | 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 $\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}$

Exercise 23.4 | Q 2 | Page 36

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

Exercise 23.4 | 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
$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4 \vec{OP}$

Exercise 23.4 | Q 4 | Page 37

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

Exercise 23.4 | 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.

Exercise 23.4 | Q 6 | Page 37

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

Exercise 23.4 [Pages 42 - 43]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.4 [Pages 42 - 43]

Exercise 23.4 | Q 1 | Page 42

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

Exercise 23.4 | 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.

Exercise 23.4 | Q 3 | Page 42

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

Exercise 23.4 | 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.

Exercise 23.4 | 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.

Exercise 23.4 | 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.

Exercise 23.4 | 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.

Exercise 23.4 | 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 $\vec{a} + 2 \vec{b} - 3 \vec{c}$.

Exercise 23.4 | 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 $\overrightarrow{AB} =$ $\vec{a}$, the coordinates of A being (4, −1).

Exercise 23.4 | Q 9 | Page 43

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

Exercise 23.4 | Q 10 | Page 43

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

Exercise 23.4 | 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 $\mu$

Exercise 23.4 | Q 12.1 | Page 43

Find the components along the coordinate axes of the position vector of the following point :

P(3, 2)

Exercise 23.4 | Q 12.2 | Page 43

Find the components along the coordinate axes of the position vector of the following point :

Q(–5, 1)

Exercise 23.4 | Q 12.3 | Page 43

Find the components along the coordinate axes of the position vector of the following point :

R(–11, –9)

Exercise 23.4 | Q 12.4 | Page 43

Find the components along the coordinate axes of the position vector of the following point :

S(4, –3)

Exercise 23.6 [Pages 48 - 49]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.6 [Pages 48 - 49]

Exercise 23.6 | Q 1 | Page 48

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

Exercise 23.6 | Q 2 | Page 48

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

Exercise 23.6 | Q 3 | Page 48

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

Exercise 23.6 | 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.

Exercise 23.6 | Q 5 | Page 49
$\text{If }\vec{a} = 3 \hat{i} - \hat{j} - 4 \hat{k} , \vec{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}\text{ and }\vec{c} = \hat{i} + 2 \hat{j} - \hat{k} ,\text{ find }\left| 3 \vec{a} - 2 \vec{b} + 4 \vec{c} \right| .$

Exercise 23.6 | Q 6 | Page 49

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

Exercise 23.6 | 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.

Exercise 23.6 | 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.

Exercise 23.6 | 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).

Exercise 23.6 | 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 2:1.

Exercise 23.6 | Q 10.2 | Page 49

Find the position vector of a point R which divides the line segment joining points:

$P \left( \hat{i} + 2 \hat{j} + \hat{k}\right) \text { and } Q \left( - \hat{i} + \hat{j} + \hat{k} \right)$ externally

Exercise 23.6 | Q 11 | Page 49

Find the position vector of the mid-point of the vector joining the points

$P \left( 2 \hat{i} - 3\hat{ j} + 4 \hat{k} \right)\text{ and } Q \left( 4 \hat{i} + \hat{j} - 2 \hat{k} \right) .$
Exercise 23.6 | Q 12 | Page 49

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

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

Exercise 23.6 | 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.

Exercise 23.6 | 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).

Exercise 23.6 | Q 15 | Page 49

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

Exercise 23.6 | 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 $2 \vec{a} - \vec{b} + 3 \vec{c .}$

Exercise 23.6 | Q 17 | Page 49

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

Exercise 23.6 | Q 18 | Page 49

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

Exercise 23.6 | Q 19 | Page 49

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

Exercise 23.7 [Pages 60 - 61]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.7 [Pages 60 - 61]

Exercise 23.7 | 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.

Exercise 23.7 | 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: $\vec{a,} \vec{b,} 3 \vec{a} - 2 \vec{b}$

Exercise 23.7 | 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: $\vec{a} + \vec{b} + \vec{c} , 4 \vec{a} + 3 \vec{b} , 10 \vec{a} + 7 \vec{b} - 2 \vec{c}$

Exercise 23.7 | 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.

Exercise 23.7 | 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.

Exercise 23.7 | Q 5 | Page 61

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

Exercise 23.7 | Q 6 | Page 61

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

Exercise 23.7 | 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.

Exercise 23.7 | Q 8 | Page 61

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

Exercise 23.7 | Q 9 | Page 61

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

Exercise 23.7 | 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.

Exercise 23.7 | 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.

Exercise 23.7 | 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).

Exercise 23.7 | 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.

Exercise 23.8 [Pages 65 - 66]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.8 [Pages 65 - 66]

Exercise 23.8 | Q 1.1 | Page 65

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

Exercise 23.8 | Q 1.2 | Page 65

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

Exercise 23.8 | 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)

Exercise 23.8 | 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)

Exercise 23.8 | 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)

Exercise 23.8 | 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)

Exercise 23.8 | 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) $\vec{a} - 2 \vec{b} + 3 \vec{c} , - 3 \vec{b} + 5 \vec{c}\text{ and }- 2 \vec{a} + 3 \vec{b} - 4 \vec{c}$
Exercise 23.8 | 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.

Exercise 23.8 | Q 5.1 | Page 65

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

Exercise 23.8 | Q 5.2 | Page 65

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

Exercise 23.8 | Q 6.1 | Page 65

Prove that the following vectors are non-coplanar:

$3 \hat{i} + \hat{j} - \hat{k} , 2 \hat{i} - \hat{j} + 7 \hat{k}\text{ and }7 \hat{i} - \hat{j} + 23 \hat{k}$
Exercise 23.8 | Q 6.2 | Page 65

Prove that the following vectors are non-coplanar:

$\hat{i} + 2 \hat{j} + 3 \hat{k} , 2 \hat{i} + \hat{j} + 3 \hat{k}\text{ and }\hat{i} + \hat{j} + \hat{k}$
Exercise 23.8 | Q 7.1 | Page 66

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

Exercise 23.8 | Q 7.2 | Page 66

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

Exercise 23.8 | 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 $\vec{d} = 2 \hat{i}-j- 3 \hat{k} , \text{ and }\text { as a linear combination of the vectors } \vec{a,} \vec{b}\text{ and }\vec{c} .$

Exercise 23.8 | 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 $l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .$

Exercise 23.8 | 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 $3 \vec{a} - 2 \vec{b} + \vec{c} - 2 \vec{d} = \vec{0} .$

Exercise 23.9 [Pages 73 - 74]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Exercise 23.9 [Pages 73 - 74]

Exercise 23.9 | Q 1 | Page 73

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

Exercise 23.9 | Q 2 | Page 73

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

Exercise 23.9 | 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.

Exercise 23.9 | 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 $\vec{r}$.

Exercise 23.9 | 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 $\vec{r}$.

Exercise 23.9 | Q 6.1 | Page 73

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

Exercise 23.9 | Q 6.2 | Page 73

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

Exercise 23.9 | Q 6.3 | Page 73

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

Exercise 23.9 | Q 7.1 | Page 73

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

Exercise 23.9 | Q 7.2 | Page 73

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

Exercise 23.9 | Q 7.3 | Page 73

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

Exercise 23.9 | Q 8 | Page 73

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

Exercise 23.9 | Q 9 | Page 73

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

Exercise 23.9 | 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 $\vec{a}$.

Exercise 23.9 | 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.

Exercise 23.9 | 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 $\vec{r}$.

Very Short Answers [Pages 75 - 77]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors Very Short Answers [Pages 75 - 77]

Very Short Answers | Q 1 | Page 75

Define "zero vector".

Very Short Answers | Q 2 | Page 75

Define unit vector.

Very Short Answers | Q 3 | Page 75

Define position vector of a point.

Very Short Answers | Q 4 | Page 75

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

Very Short Answers | 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.

Very Short Answers | Q 6 | Page 75

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

Very Short Answers | 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 $\vec{a} + \vec{b} + \vec{c} .$

Very Short Answers | 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 $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} .$

Very Short Answers | 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 $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} .$

Very Short Answers | Q 10 | Page 75

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

Very Short Answers | Q 11 | Page 75

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

Very Short Answers | Q 12 | Page 75

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

Very Short Answers | Q 13 | Page 75

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

Very Short Answers | 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 $\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} .$

Very Short Answers | Q 15 | Page 75

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

Very Short Answers | 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 $\vec{a} + \vec{b} + \vec{c} .$

Very Short Answers | Q 17 | Page 75

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

Very Short Answers | 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.

Very Short Answers | 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.

Very Short Answers | Q 20 | Page 76

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

Very Short Answers | 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 $\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\overrightarrow{b} = - \hat{i} + \hat{j} + \hat{k} .$

Very Short Answers | Q 22 | Page 76

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

Very Short Answers | Q 23 | Page 76

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

Very Short Answers | Q 24 | Page 76

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

Very Short Answers | 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.

Very Short Answers | Q 26 | Page 76

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

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

A unit vector $\overrightarrow{r}$ 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.

Very Short Answers | Q 29 | Page 76

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

Very Short Answers | Q 30 | Page 76

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

Very Short Answers | Q 31 | Page 76

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

Very Short Answers | 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).

Very Short Answers | Q 33 | Page 76

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

Very Short Answers | 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?

Very Short Answers | Q 35 | Page 76

Write two different vectors having same magnitude.

Very Short Answers | Q 36 | Page 76

Write two different vectors having same direction.

Very Short Answers | 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.

Very Short Answers | Q 38 | Page 76

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

Very Short Answers | Q 39 | Page 76

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

Very Short Answers | 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?

Very Short Answers | Q 41 | Page 76

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

Very Short Answers | Q 42 | Page 76

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

Very Short Answers | Q 43 | Page 76

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

Very Short Answers | Q 44 | Page 77

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

Very Short Answers | Q 45 | Page 77

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

Very Short Answers | 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.

Very Short Answers | Q 47 | Page 77

Find a vector $\overrightarrow{a}$ 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.

Very Short Answers | Q 48 | Page 77

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

Very Short Answers | 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.

Very Short Answers | Q 50 | Page 77

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

Very Short Answers | Q 51 | Page 77

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

Very Short Answers | Q 52 | Page 77

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

MCQ [Pages 78 - 79]

### RD Sharma solutions for Class 12 Maths Chapter 23 Algebra of Vectors MCQ [Pages 78 - 79]

MCQ | 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

MCQ | 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

• $\vec{a} + \vec{b}$

• $\vec{a} - \vec{b}$

• $\vec{b} - \vec{a}$

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

MCQ | 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

MCQ | 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

MCQ | 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

MCQ | Q 6 | Page 78

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

• $2 \overrightarrow{OG}$

• $4 \overrightarrow{OG}$

• $5 \overrightarrow{OG}$

• $3 \overrightarrow{OG}$
MCQ | 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

MCQ | Q 8 | Page 78

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

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

• $\vec{b} + \vec{c}$

• $\vec{a} + 2 \vec{b} + 2 \vec{c}$

MCQ | 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

MCQ | Q 10 | Page 78

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

• 2$\overrightarrow{OO}$

• $O \overrightarrow{O'}$
• $\overrightarrow{OO'}$

• $2 \overrightarrow{O'O}$
MCQ | Q 11 | Page 78

If $\vec{a}$, $\vec{b}$, $\vec{c}$ 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

MCQ | Q 12 | Page 79

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

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

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

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

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

• $2 \overrightarrow{AB}$

• $\vec{0}$
• $3 \overrightarrow{AB}$

• $4 \overrightarrow{AB}$
MCQ | 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

MCQ | 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)

MCQ | Q 16 | Page 79

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

• $4 \overrightarrow{AB}$

• $3 \overrightarrow{AB}$
• $2 \overrightarrow{AB}$

• $\overrightarrow{AB}$
MCQ | Q 17 | Page 79

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

• $\left( \vec{a} + \vec{b} \right)$

• $\left( \vec{a} - \vec{b} \right)$

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

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

MCQ | 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

MCQ | Q 19 | Page 79

In Figure, which of the following is not true?

• $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}$

• $\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AC} = \vec{0}$

• $\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{CA} = \vec{0}$

• $\overrightarrow{AB} - \overrightarrow{CB} + \overrightarrow{CA} = \vec{0}$

## Solutions for Chapter 23: Algebra of Vectors

Exercise 23.1Exercise 23.2Exercise 23.3Exercise 23.4Exercise 23.4Exercise 23.6Exercise 23.7Exercise 23.8Exercise 23.9Very Short AnswersMCQ ## RD Sharma solutions for Class 12 Maths chapter 23 - Algebra of Vectors

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Concepts covered in Class 12 Maths chapter 23 Algebra of Vectors are 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, Multiplication of a Vector by a Scalar, Addition of Vectors, Vectors and Their Types, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Components of 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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