Question

If $|a\times b{|}^2+|a.b{|}^2=144$ and $\left|a\right|=4$, then $\left|b\right|$ is equal to?

• A. $16$
• B. $8$
• C. $3$
• D. $12$

Answer 1

The correct option is C

$3$

Explanation for the correct option:

Step 1. Find the value of $\left|b\right|$:

Given,

$|a\times b{|}^2+|a.b{|}^2=144$ ….(1)

Step 2. Let the angle between $a$ and $b$ is $\theta$, then equation (1) becomes:

$\left|a{|}^2\right|b{|}^2{\sin }^2\theta +|a{|}^2|b{|}^2{\cos }^2\theta =144$

$\Rightarrow$ $\left|a{|}^2\right|b{|}^2({\sin }^2\theta +{\cos }^2\theta )=144$

$\Rightarrow$ $\left|a{|}^2\right|b{|}^2=144$ $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

$\Rightarrow$ $\left|a\right|\left|b\right|=12$

$\Rightarrow$ $4\left|b\right|=12$ $\left[\because \left|a\right|=4\right]$

$\mathbf{\because }\mathbf{\left|}\mathit{b}\mathbf{\right|}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}$

Hence, Option ‘C’ is Correct.