Let a → = ( 2 i ^ + 3 j ^ + 2 k ^ ) and b → = ( i ^ + 2 j ^ + k ^ ) . Find the projection of a → on b → .

Given, a → = ( 2 i ^ + 3 j ^ + 2 k ^ ) b → = ( i ^ + 2 j ^ + k ^ ) So, | b → | = 1 2 + 2 2 + 1 2 = 1 + 4 + 1 = 6 Projection of a → on b → is = a → · b → b → = ( 2 i ^ + 3 j ^ + 2 k ^ ) · i ^ + 2 j ^ + k ^ 6 = 2 + 6 + 2 6 = 10 6 So, the projection of a → on b → is 10 6 .

Let a → = ( 2 i ^ + 3 j ^ + 2 k ^ ) and b → = ( i ^ + 2 j ^ + k ^ ) . Find the projection of b → on a → .

Given, a → = ( 2 i ^ + 3 j ^ + 2 k ^ ) b → = ( i ^ + 2 j ^ + k ^ ) | a → | = 2 2 + 3 2 + 2 2 = 4 + 9 + 4 = 17 Projection of b → on a → is = b → · a → a → = ( i ^ + 2 j ^ + k ^ ) · 2 i ^ + 3 j → + 2 k ^ 17 = 2 + 6 + 2 17 = 10 17 So, the projection of b → on a → is 10 17 .

Find the angle between the vectors a → and b → , when a → = i ^ - 2 j ^ + 3 k ^ and b → = 3 i ^ - 2 j ^ + k ^ .

Given, a → = i ^ - 2 j ^ + 3 k ^ and b → = 3 i ^ - 2 j ^ + k ^ a → = ( i ^ - 2 j ^ + 3 k ^ ) b → = ( 3 i ^ - 2 j ^ + k ^ ) So, a → = 1 2 + ( - 2 ) 2 + 3 2 = 1 + 4 + 9 = 14 | b → | = 3 2 + ( - 2 ) 2 + 1 2 = 9 + 4 + 1 = 14 We know that a → · b → = | a → | | b → | cos θ ⇒ cos θ = a → · b → | a → | | b → | = ( i ^ - 2 j ^ + 3 k ^ ) · ( 3 i ^ - 2 j ^ + k ^ ) 14 14 = 3 + 4 + 3 196 ⇒ cos θ = 10 14 ⇒ θ = cos - 1 5 7 Which is the required angle between the given two vectors.

Find the angle between the vectors a → and b → , when a → = 3 i ^ + j ^ + 2 k ^ and b → = 2 i ^ - 2 j ^ + 4 k ^ .

Given, a → = 3 i ^ + j ^ + 2 k ^ and b → = 2 i ^ - 2 j ^ + 4 k ^ So, a → = 3 2 + ( 1 ) 2 + 2 2 = 9 + 1 + 4 = 14 | b → | = 2 2 + ( - 2 ) 2 + 4 2 = 4 + 4 + 16 = 24 We know that a → · b → = | a → | | b → | cos θ ⇒ cos θ = a → · b → | a → | | b → | = ( 3 i ^ + j ^ + 2 k ^ ) · ( 2 i ^ - 2 j ^ + 4 k ^ ) 14 24 = ( 6 - 2 + 8 ) 336 ⇒ cos θ = 12 336 ⇒ cos θ = 144 336 ⇒ θ = cos - 1 3 7 Which is the required angle between the given two vectors.

If a → = i ^ + 2 j ^ - 3 k ^ and b → = 3 i ^ - j ^ + 2 k ^ then calculate the angle between 2 a → + b → and a → + 2 b → .

Given, a → = i ^ + 2 j ^ - 3 k ^ b → = 3 i ^ - j ^ + 2 k ^ ⇒ a → + 2 b → = i ^ + 2 j ^ - 3 k ^ + 2 3 i ^ - j ^ + 2 k ^ = 7 i ^ + k ^ ⇒ 2 a → + b → = 2 i ^ + 2 j ^ - 3 k ^ + 3 i ^ - j ^ + 2 k ^ = 5 i ^ + 3 j - 4 k ^ So, a → + 2 b → = 7 2 + 1 2 = 50 2 a → + b → = 5 2 + 3 2 + - 4 2 = 50 We know that dot product of two vectors is given by l → · m → = | l → | | m → | cos θ So, cos θ = a → + 2 b → · 2 a → + b → a → + 2 b → 2 a → + b → = 7 i ^ + k ^ · 5 i ^ + 3 j - 4 k ^ 50 50 = 35 - 4 50 ⇒ cos θ = 31 50 ⇒ θ = cos - 1 31 50 Which is the required angle between the given two vectors.

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Vector Algebra>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE

R S Aggarwal Solutions for Exercise 2: EXERCISE

Author:R S Aggarwal

R S Aggarwal Mathematics Solutions for Exercise - R S Aggarwal Solutions for Exercise 2: EXERCISE

Attempt the free practice questions from Exercise 2: EXERCISE with hints and solutions to strengthen your understanding. SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12 solutions are prepared by Experienced Embibe Experts.

Questions from R S Aggarwal Solutions for Exercise 2: EXERCISE with Hints & Solutions

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 7

Let $\vec{a} = (2 \hat{i} + 3 \hat{j} + 2 \hat{k})$ and $\vec{b} = (\hat{i} + 2 \hat{j} + \hat{k})$ . Find the projection of $\vec{a}$ on $\vec{b}$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 7

Let $\vec{a} = (2 \hat{i} + 3 \hat{j} + 2 \hat{k})$ and $\vec{b} = (\hat{i} + 2 \hat{j} + \hat{k})$ . Find the projection of $\vec{b}$ on $\vec{a}$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 11

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

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 11

Find the angle between the vectors $\vec{a}$ and $\vec{b}$ , when $\vec{a} = 3 \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 12

If $\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}$ then calculate the angle between $2 \vec{a} + \vec{b}$ and $\vec{a} + 2 \vec{b}$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 28

Find the value of $λ$ for which the vectors $(2 \hat{i} + λ \hat{j} + 3 \hat{k})$ and $(3 \hat{i} + 2 \hat{j} - 4 \hat{k})$ are perpendicular to each other.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 31

If the position vectors of the vertices $A, B$ and $C$ of a $△ A B C$ be $(1, 2, 3), (- 1, 0, 0)$ and $(0, 1, 2)$ respectively then find $∠ A B C$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 23 - Scalars or Dot, Product of Vectors>EXERCISE>Q 32

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

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