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# RD Sharma solutions for Class 12 Mathematics chapter 24 - Scalar Or Dot Product

## Chapter 24: Scalar Or Dot Product

#### Chapter 24: Scalar Or Dot Product Exercise 24.1 solutions [Pages 29 - 33]

Ex. 24.1 | Q 1.1 | Page 29

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

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

Ex. 24.1 | Q 1.2 | Page 29

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

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

Ex. 24.1 | Q 1.3 | Page 29

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

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

Ex. 24.1 | Q 2.1 | Page 30

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

Ex. 24.1 | Q 2.2 | Page 30

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

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

Ex. 24.1 | Q 2.3 | Page 30

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

$\vec{a} = 2 \hat{i} + 3 \hat{j} + 4\hat{k} \text{ and } \vec{b} = 3 \hat{i} - 2 \hat{j} +\lambda \hat{k}$

Ex. 24.1 | Q 2.4 | Page 30

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

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

Ex. 24.1 | 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 $\vec{a} \text{ and } \vec{b} .$

Ex. 24.1 | Q 4 | Page 30

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

Ex. 24.1 | Q 5.1 | Page 30

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

Ex. 24.1 | Q 5.2 | Page 30

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

Ex. 24.1 | Q 5.3 | Page 30

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

Ex. 24.1 | Q 5.4 | Page 30

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

Ex. 24.1 | Q 5.5 | Page 30

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

Ex. 24.1 | 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.

Ex. 24.1 | 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.

Ex. 24.1 | 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.

Ex. 24.1 | Q 8.1 | Page 30

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

Ex. 24.1 | Q 8.2 | Page 30

If  $\hat{ a } \text{ and } \hat{b }$ are unit vectors inclined at an angle θ, prove that

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

Ex. 24.1 | Q 9 | Page 30

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

Ex. 24.1 | Q 10 | Page 30

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

Ex. 24.1 | Q 11 | Page 30

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

Ex. 24.1 | Q 12 | Page 30

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

Ex. 24.1 | Q 13 | Page 30

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

Ex. 24.1 | Q 14 | Page 30

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

Ex. 24.1 | 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 $\lambda \vec{b} + \vec{c}$

Ex. 24.1 | 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.

Ex. 24.1 | 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 $\vec{\alpha}$

Ex. 24.1 | 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.

Ex. 24.1 | 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.

Ex. 24.1 | 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.

Ex. 24.1 | 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).

Ex. 24.1 | 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.

Ex. 24.1 | 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.

Ex. 24.1 | 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

Ex. 24.1 | 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

Ex. 24.1 | Q 26 | Page 31

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

Ex. 24.1 | 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.

Ex. 24.1 | Q 28 | Page 31

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

Ex. 24.1 | 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 $\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .$

Ex. 24.1 | Q 30.1 | Page 31

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

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

Ex. 24.1 | Q 30.2 | Page 31

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

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

Ex. 24.1 | Q 31.1 | Page 31

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

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

Ex. 24.1 | Q 31.2 | Page 31

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

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

Ex. 24.1 | Q 31.3 | Page 31

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

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

Ex. 24.1 | Q 32.1 | Page 31

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

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

Ex. 24.1 | Q 32.2 | Page 31

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

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

Ex. 24.1 | Q 32.3 | Page 31

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

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

Ex. 24.1 | Q 33.1 | Page 31

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

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

Ex. 24.1 | Q 33.2 | Page 31

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

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

Ex. 24.1 | 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 $\vec{b}$

Ex. 24.1 | 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 $\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .$

Ex. 24.1 | 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 $2 \hat{i} + 4 \hat{j} - 2 \hat{k} .$

Ex. 24.1 | 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 $\hat{i} + \hat{j} + \hat{k} .$

Ex. 24.1 | 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 $\vec{a} - \vec{b}$

Ex. 24.1 | 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 $\vec{b}$ ?

Ex. 24.1 | 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 $\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}$

Ex. 24.1 | Q 41 | Page 32

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

Ex. 24.1 | 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.

Ex. 24.1 | 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 $\vec{b} \text{ and } \vec{c} .$

Ex. 24.1 | 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  $\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .$

Ex. 24.1 | 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 $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .$

Ex. 24.1 | 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}\$  is acute and the angle between $\vec{b} \text{ and } \vec{c}$ is obtuse.

Ex. 24.1 | Q 47 | 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.

Ex. 24.1 | Q 48 | Page 32

If  $\vec{a} \text{ and } \vec{b}$ are two non-collinear unit vectors such that $\left| \vec{a} + \vec{b} \right| = \sqrt{3},$ find $\left( 2 \vec{a} - 5 \vec{b} \right) \cdot \left( 3 \vec{a} + \vec{b} \right) .$

Ex. 24.1 | 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 $\vec{a}$

#### Chapter 24: Scalar Or Dot Product Exercise 24.2 solutions [Page 46]

Ex. 24.2 | Q 1 | Page 46

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

Ex. 24.2 | Q 2 | Page 46

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

Ex. 24.2 | 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.

Ex. 24.2 | 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.

Ex. 24.2 | Q 5 | Page 46

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

Ex. 24.2 | Q 6 | Page 46

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

Ex. 24.2 | Q 7 | Page 46

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

Ex. 24.2 | Q 8 | Page 46

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

Ex. 24.2 | Q 9 | Page 46

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

Ex. 24.2 | 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.

#### Chapter 24: Scalar Or Dot Product Exercise very short answer solutions [Pages 46 - 48]

very short answer | Q 1 | Page 46

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

very short answer | 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 $\vec{a} \text{ on } \vec{b}$

very short answer | Q 3 | Page 46

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

very short answer | 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

very short answer | 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.

very short answer | Q 6 | Page 46

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

very short answer | 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 $\vec{a} \text{ and } \vec{b}$

very short answer | 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.

very short answer | 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.

very short answer | 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.

very short answer | Q 11 | Page 47

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

very short answer | Q 12 | Page 47

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

very short answer | 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 $\left| \hat{a} - \hat{b} \right| .$

very short answer | Q 14 | Page 47

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

very short answer | Q 15 | Page 47

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

very short answer | Q 16 | Page 47

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

very short answer | Q 17 | Page 47

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

very short answer | Q 18 | Page 47

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

very short answer | 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.

very short answer | 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.

very short answer | Q 21 | Page 47

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

very short answer | Q 22 | Page 47

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

very short answer | Q 23 | Page 47

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

very short answer | Q 24 | Page 47

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

very short answer | Q 25 | Page 47

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

very short answer | Q 26 | Page 47

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

very short answer | 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?

very short answer | Q 28 | Page 47

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

very short answer | Q 29 | Page 47

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

very short answer | 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.

very short answer | 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 $\vec{b} \text{ on } \vec{a}$

very short answer | Q 32 | Page 48

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

very short answer | Q 33 | Page 48

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

very short answer | 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.

very short answer | 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?

very short answer | Q 36 | Page 48

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

very short answer | 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.

very short answer | Q 38 | Page 48

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

very short answer | 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 $\left| \vec{b} \right|$

very short answer | 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 $\vec{b}$

very short answer | 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 $\vec{b}$

very short answer | 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.

#### Chapter 24: Scalar Or Dot Product Exercise MCQ solutions [Pages 49 - 51]

MCQ | 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

•  $\cos \theta = \frac{4}{5}$

•  $\sin \theta = \frac{1}{\sqrt{2}}$

•  $\cos \theta = - \frac{4}{5}$

•  $\cos \theta = - \frac{3}{5}$

MCQ | 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 $\vec{a} =$

• $\vec{0}$

•  $\hat{i}$

•   $\hat{j}$

• $\hat{i} + \hat{j} + \hat{k}$

MCQ | 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

•  $\frac{\pi}{6}$

•  $\frac{2\pi}{3}$

•  $\frac{5\pi}{3}$

•  $\frac{\pi}{3}$

MCQ | 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

• $\alpha = \frac{\pi}{4}$

• $\alpha = \frac{\pi}{3}$

•  $\alpha = \frac{2\pi}{3}$

•  $\alpha = \frac{\pi}{2}$

MCQ | Q 5 | Page 49

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 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

•  $\frac{5}{\sqrt{162}}$

• $\frac{4}{\sqrt{162}}$

•  $- \frac{5}{\sqrt{162}}$

• $\frac{11}{\sqrt{162}}$

MCQ | 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?

•  $\sqrt{3}$

• $\sqrt{3}/2$

•  $1/\sqrt{2}$

•  −1/2

MCQ | 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 circle

• an ellipse

• a hyperbola

•  None of these

MCQ | Q 9 | Page 49

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

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

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

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

•  None of these

MCQ | 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?

• 15

•  $\sqrt{113}$

•  $\sqrt{593}$

•  $\sqrt{369}$

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

MCQ | 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

MCQ | 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 $\frac{\pi}{6}$  are

• (a) $x > \frac{1}{2} or x < 0$

• (b) $0 < x < \frac{1}{2}$

• (c) $\frac{1}{2} < x < 15$

• (d) ϕ

MCQ | 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

MCQ | 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) $\frac{1}{7}$

MCQ | Q 16 | Page 50

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

• (a) 1

• (b) 0

• (c) 2

• (d) −1

• (e) −2

MCQ | 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

MCQ | 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

MCQ | 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

MCQ | Q 20 | Page 50

If $\vec{a} \text{ and } \vec{b}$ 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

MCQ | 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)

MCQ | 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) $\frac{2\pi}{3} < \theta < \pi$

MCQ | 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

MCQ | 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) $\frac{\vec{b}}{\left| \vec{b} \right|}$

MCQ | 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)  $\frac{\pi}{3}$

MCQ | 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) $\frac{2\pi}{3}$

## RD Sharma solutions for Class 12 Mathematics chapter 24 - Scalar Or Dot Product

RD Sharma solutions for Class 12 Maths chapter 24 (Scalar Or Dot Product) 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 24 Scalar Or Dot Product are Introduction of Product of Two Vectors, Projection of a Vector on a Line, Vectors Examples and Solutions, Vector Joining Two Points, Section formula, Components of a Vector, Types of Vectors, Basic Concepts of Vector Algebra, Magnitude and Direction of a Vector, Introduction of Vector, 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.

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