Find the value of the expression | a → × b → | 2 + a → · b → 2 .

Given, | a → × b → | 2 + | a → · b → | 2 We know that, cross product of two vectors is a → × b → = | a → | | b → | sinθ and dot product of two vectors is a → . b → = | a → | | b → | cosθ , where θ is the angle between a → , b → . Now, | a → × b → | 2 + | a → · b → | 2 = | a → | 2 | b → | 2 sin 2 θ + | a → | 2 | b → | 2 cos 2 θ = | a → | 2 b → 2 cos 2 θ + sin 2 θ = | a → | 2 b → 2 1 ∵ cos 2 θ + sin 2 θ = 1 Therefore, | a → × b → | 2 + | a → · b → | 2 = | a → | 2 b → 2 .

If a → is any non-zero vector, then a → · i → i ^ + a → · j → j ^ + a → · k → k ^ equals _______.

Given that, a → is a non-zero vector Let a → = a 1 i ^ + a 2 j ^ + a 3 k ^ We know that, from properties of scalar product i ^ . i ^ = j ^ . j ^ = k ^ . k ^ = 1 and i ^ . j ^ = j ^ . k ^ = k ^ . i ^ = 0 Here, a → · i ^ = a 1 i ^ + a 2 j ^ + a 3 k ^ . i ^ = a 1 i ^ . i ^ + a 2 j ^ . i ^ + a 3 k ^ . i ^ = a 1 ( 1 ) + 0 + 0 = a 1 a → · j ^ = a 1 i ^ + a 2 j ^ + a 3 k ^ . j ^ = a 1 i ^ . j ^ + a 2 j ^ . j ^ + a 3 k ^ . j ^ = ( 0 ) + a 2 ( 1 ) + 0 = a 2 a → · k ^ = a 1 i ^ + a 2 j ^ + a 3 k ^ . k ^ = a 1 i ^ . k ^ + a 2 j ^ . k ^ + a 3 k ^ . k ^ = 0 + 0 + a 3 ( 1 ) = a 3 Now, a → · i → i ^ + a → · j → j ^ + a → · k → k ^ = a 1 i ^ + a 2 j ^ + a 3 k ^ = a → Therefore, the value of a → · i → i ^ + a → · j → j ^ + a → · k → k ^ = a → .

E M B I B E

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE

NCERT Solutions for Chapter: Vector Algebra, Exercise 1: EXERCISE

Author:NCERT

NCERT Mathematics Solutions for Exercise - NCERT Solutions for Chapter: Vector Algebra, Exercise 1: EXERCISE

Attempt the practice questions on Chapter 10: Vector Algebra, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. NCERT Exemplar Mathematics - Class 12 solutions are prepared by Experienced Embibe Experts.

Questions from NCERT Solutions for Chapter: Vector Algebra, Exercise 1: EXERCISE with Hints & Solutions

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 2

If $\vec{a} = \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and the unit vector in the direction of $6 \vec{b}$ is $\frac{m \hat{ı} + n \hat{ȷ} + p \hat{k}}{3}$ , then the value of $m + n + p$ is

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 2

If $\vec{a} = \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - 2 \hat{k}$ , then the unit vector in the direction of $2 \vec{a} - \vec{b}$ is $m (y \hat{j} + z \hat{k})$ , find $z$ .

MEDIUM

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 16

If $\vec{a}, \vec{b}, \vec{c}$ determine the vertices of a triangle, show that $\frac{1}{2} [\vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{a} \times \vec{b}]$ gives the vector area of the triangle. Hence deduce the condition that the three points $\vec{a}, \vec{b}, \vec{c}$ are collinear. Also find the unit vector normal to the plane of the triangle.

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 38

Find the value of the expression $| \vec{a} \times \vec{b} |^{2} + {|\vec{a} \cdot \vec{b}|}^{2}$ .

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 40

If $\vec{a}$ is any non-zero vector, then $(\vec{a} \cdot \vec{i}) \hat{i} + (\vec{a} \cdot \vec{j}) \hat{j} + (\vec{a} \cdot \vec{k}) \hat{k}$ equals _______.

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 43

If $| \vec{a} + \vec{b} | = | \vec{a} - \vec{b} |$ , then the vectors $\vec{a}$ and $\vec{b}$ are orthogonal.

EASY

12th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 12>Chapter 10 - Vector Algebra>EXERCISE>Q 44

The formula of $(\vec{a} + \vec{b})^{2} = {\vec{a}}^{2} + {\vec{b}}^{2} + 2 \vec{a} \cdot \vec{b}$ is valid for non-zero vectors $\vec{a}$ and $\vec{b}$ .