Find the sum of the following vectors: a → = i ^ - 2 j ^ , b → = 2 i ^ - 3 j ^ , c → = 2 i ^ + 3 k ^ .

Given, a → = i ^ - 2 j ^ , b → = 2 i ^ - 3 j ^ , c → = 2 i ^ + 3 k ^ . So, Sum of the three vectors = a → + b → + c → = i ^ - 2 j ^ + 2 i ^ - 3 j ^ + 2 i ^ + 3 k ^ = 5 i ^ - 5 j ^ + 3 k ^

Write a unit vector in the direction of the sum of the vectors a → = 2 i ^ + 2 j ^ - 5 k ^ and b → = 2 i ^ + j ^ - 7 k ^ .

We have, a → = 2 i ^ + 2 j ^ - 5 k ^ and b → = 2 i ^ + j ^ - 7 k ^ . ∴ a → + b → = 2 i ^ + 2 j ^ - 5 k ^ + 2 i ^ + j ^ - 7 k ^ = 4 i ^ + 3 j ^ - 12 k ^ ∴ Unit vector along a → + b → = a → + b → a → + b → = 4 i ^ + 3 j ^ - 12 k ^ 4 i ^ + 3 j ^ - 12 k ^ = 4 i ^ + 3 j ^ - 12 k ^ 4 2 + 3 2 + - 12 2 = 4 i ^ + 3 j ^ - 12 k ^ 169 = 4 i ^ + 3 j ^ - 12 k ^ 13 Which is the required unit vector along a → + b → .

Find a vector a → of magnitude 5 2 , making an angle of π 4 with x -axis, π 2 with y -axis and an acute angle θ with z -axis.

It is given that vector a → makes an angle of π 4 with x -axis, π 2 with y -axis and an acute angle θ with z -axis. ∴ l = cos π 4 = 1 2 , m = cos π 2 = 0 , n = cos θ Now, l 2 + m 2 + n 2 = 1 ⇒ 1 2 + 0 + cos 2 θ = 1 ⇒ cos 2 θ = 1 - 1 2 = 1 2 ⇒ cos θ = 1 2 ( θ is acute) We know that a → = | a → | ( l i ^ + m j ^ + n k ^ ) ⇒ a → = 5 2 1 2 i ^ + 0 j ^ + 1 2 k ^ ⇒ a → = 5 i ^ + 5 k ^

Write a unit vector in the direction of P Q → , where P and Q are the points ( 1 , 3 , 0 ) and ( 4 , 5 , 6 ) respectively.

P ( 1 , 3 , 0 ) and Q ( 4 , 5 , 6 ) are the given points. ∴ P Q → = ( 4 i ^ + 5 j ^ + 6 k ^ ) - ( i ^ + 3 j ^ + 0 k ^ ) = 3 i ^ + 2 j ^ + 6 k ^ ∴ Unit vector in the direction of P Q → = P Q → | P Q → | = 3 i ^ + 2 j ^ + 6 k ^ 3 i ^ + 2 j ^ + 6 k ^ = 3 i ^ + 2 j ^ + 6 k ^ 3 2 + 2 2 + 6 2 = 3 i ^ + 2 j ^ + 6 k ^ 49 = 3 i ^ + 2 j ^ + 6 k ^ 7

Find a vector in the direction of vector 2 i ^ - 3 j ^ + 6 k ^ which has magnitude 21 units.

Let a → = 2 i ^ - 3 j ^ + 6 k ^ To find: A vector of 21 units in the direction of the vector a → . We know that, unit vector along a → is a ^ = a → a → So, a ^ = 2 i ^ - 3 j ^ + 6 k ^ 2 2 + - 3 2 + 6 2 = 2 i ^ - 3 j ^ + 6 k ^ 49 = 2 i ^ - 3 j ^ + 6 k ^ 7 So, vector of 21 units is 21 a ^ = 21 2 i ^ - 3 j ^ + 6 k ^ 7 = 6 i ^ - 9 j ^ + 18 k ^

If a → = 4 and - 3 ≤ λ ≤ 2 , then write the range of | λ a ^ | .

Given, a → = 4 It is given that - 3 ≤ λ ≤ 2 ⇒ - 3 × | a → | ≤ λ | a → | ≤ 2 × | a → | ⇒ - 3 × 4 ≤ | λ a → | ≤ 2 × 4 ( k | a → | = | k a → | , k is scalar) ⇒ - 12 ≤ | λ a → | ≤ 8 Thus, the range of | λ a → | is [ - 12 , 8 ] .

In a triangle O A C , if B is the mid-point of side A C and O A → = a → , O B → = b → , then what is O C → ?

In △ O A C , O A → = a → and O B → = b → It is given that B is the mid-point of A C So, B divides A C in the ratio 1 : 1 ∴ Position vector of B = 1 Position vector of A + 1 Position vector of C 1 + 1 ⇒ O B → = O A → + O C → 2 ⇒ b → = a → + O C → 2 ⇒ O C → = 2 b → - a →

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)

R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 10: VERY SHORT ANSWER QUESTIONS (VSAQs)

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 10: VERY SHORT ANSWER QUESTIONS (VSAQs)

Attempt the free practice questions on Chapter 4: Algebra of Vectors, Exercise 10: VERY SHORT ANSWER QUESTIONS (VSAQs) with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-2 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 10: VERY SHORT ANSWER QUESTIONS (VSAQs) with Hints & Solutions

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 42

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

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 45

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 $\vec{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}$ .

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 47

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

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 48

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

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 49

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

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 50

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

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 51

In a triangle $O A C$ , if $B$ is the mid-point of side $A C$ and $\vec{O A} = \vec{a}, \vec{O B} = \vec{b}$ , then what is $\vec{O C}$ ?

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 52

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

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