Question

2 points
Vector A = 4î + 4j -4k and vector B =3î+ j + 4k , then angle between vectors A and B is

Answer 1

$\large\underline{\sf{Solution-}}$

Given vectors are

$\rm :\longmapsto\:\vec{a} = 4\hat{i} + 4\hat{j} - 4\hat{k}$

and

$\rm :\longmapsto\:\vec{b} = 3\hat{i} + \hat{j} + 4\hat{k}$

Now, we know, angle between two vectors is given by

$\boxed{ \tt{ \: cos \theta \: = \frac{\vec{a} .\vec{b} }{ |\vec{a} | \: |\vec{b} | } \: \: }}$

So, Let evaluate

$\rm :\longmapsto\: |\vec{a} |$

$\rm \: = \: |4\hat{i} + 4\hat{j} - 4\hat{k} |$

$\rm \: = \: \sqrt{ {4}^{2} + {4}^{2} + {( - 4)}^{2} }$

$\rm \: = \: \sqrt{16 + 16 + 16}$

$\rm \: = \: \sqrt{16 \times 3}$

$\rm \: = \:4 \sqrt{3}$

$\bf\implies \:\boxed{ \tt{ \: |\vec{a} | = 4 \sqrt{3} \: \: }}$

Now, Consider,

$\rm :\longmapsto\: |\vec{b} |$

$\rm \: = \: |3\hat{i} + \hat{j} + 4\hat{k} |$

$\rm \: = \: \sqrt{ {3}^{2} + {1}^{2} + {4}^{2} }$

$\rm \: = \: \sqrt{9 + 1 + 16}$

$\rm \: = \: \sqrt{26}$

$\bf\implies \:\boxed{ \tt{ \: |\vec{b} | = \sqrt{26} \: \: }}$

Now, Consider

$\rm :\longmapsto\:\vec{a} .\vec{b}$

$\rm \: = \:(4\hat{i} + 4\hat{j} - 4\hat{k} ).(3\hat{i} + \hat{j} + 4\hat{k} )$

$\rm \: = \:12 + 4 - 16$

$\rm \: = \:16 - 16$

$\rm \: = \:0$

$\bf\implies \:\boxed{ \tt{ \: \vec{a} .\vec{b} = 0 \: \: }}$

Now,

$\rm :\longmapsto\:cos\theta \: = \: \dfrac{\vec{a} .\vec{b} }{ |\vec{a} | \: |\vec{b} | }$

$\rm :\longmapsto\:cos\theta \: = \: \dfrac{0 }{ 4 \sqrt{3} \times \sqrt{26} }$

$\rm :\longmapsto\:cos\theta \: = \: 0$

$\bf\implies \:\theta \: = \: \dfrac{\pi}{ 2}$

Additional Information :-

$\boxed{ \tt{ \: \vec{a} .\vec{b} = \vec{b} .\vec{a} \: }}$

$\boxed{ \tt{ \: \vec{a} .\vec{b} = |\vec{a} | |\vec{b} | cos\theta \: \: \: }}$

$\boxed{ \tt{ \: |\vec{a} \times \vec{b} | = |\vec{a} | |\vec{b} | sin\theta \: \: }}$

$\boxed{ \tt{ \: \vec{a} \times \vec{b} \: = - \: \vec{b} \times \vec{a} \: \: }}$

$\boxed{ \tt{ \: \vec{a} \times \vec{b} = 0 \: \implies \: \vec{a} \parallel \: \vec{b} \: \: }}$

$\boxed{ \tt{ \: \vec{a}.\vec{b} = 0 \: \implies \: \vec{a} \perp\: \vec{b} \: \: }}$

Answer 2

Step-by-step explanation:

A.B = 12 + 4 - 16 = 0

So, A is perpendicular to B

So, angle is 90