Question

If $\left|a\right|=5,\left|b\right|=6$and $a.b=–25,$then $|a\times b|$is equal to

• A. $25$
• B. $6\sqrt{11}$
• C. $11\sqrt{5}$
• D. $11\sqrt{6}$
• E. $5\sqrt{1}1$

Answer 1

The correct option is E

$5\sqrt{1}1$

The explanation of the correct option :

Step1. Find the angle:

Let the angle be $\theta$.

Given,

$\left|a\right|=5,\left|b\right|=6$ and $a.b=–25$

We know that

$\cos \theta =\frac{a.b}{\left|a\right|\times \left|b\right|}$

$\Rightarrow \cos \theta =\frac{-25}{5\times 6}$

$\Rightarrow \cos \theta =\frac{-5}{6}$

Also, we know that

$si{n}^2\theta +{\cos }^2\theta =1$

$\begin{array}{rcl}\therefore \sin \theta & =& \sqrt{1-{\cos }^2\theta }\\ & =& \sqrt{1-\frac{25}{36}}\\ & =& \sqrt{\frac{11}{36}}\end{array}$

Step2. Finding the value of $|a\times b|$:

$\therefore |a\times b|=\left|a\right|\left|b\right|\sin \theta$

$\begin{array}{rcl}\therefore |a\times b|& =& 5\times 6\times \sqrt{\frac{11}{36}}\\ & \mathbf{=}& \mathbf{5}\sqrt{\mathbf{11}}\end{array}$

Therefore, $|a\times b|=5\sqrt{11}$

Hence, option (E) is the correct answer.