Question

If $| \vec{a} | = 13 , |\vec{b} | = 5 $ and $\vec{a} . \vec{b} = 30$ , then $|\vec{a} \times \vec{b} | $ is equal to

• A. 30
• B. $\frac{30}{25} \sqrt{233} $
• C. $\frac{30}{33} \sqrt{193} $
• D. $\frac{65}{23} \sqrt{493} $
• E. $\frac{65}{13} \sqrt{133} $

Answer 1

Answer: (E)

Given that,
$|\vec{a}|=13,|\vec{b}|=5$ and $\vec{a} \cdot \vec{b}=30$
$\because \, \vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$
$\Rightarrow \, 30=13.5 \cos \theta$
$\Rightarrow \, \cos \theta=\frac{30}{13.5}=\frac{6}{13}$
$\Rightarrow \, \sin ^{2} \theta=1-\frac{36}{169}$
$\Rightarrow \, \sin ^{2} \theta=\frac{133}{169}$
$\Rightarrow \, \sin \theta=\frac{\sqrt{133}}{13}$
$\Rightarrow |\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta$
$= 13.5 \cdot \frac{\sqrt{133}}{13}$
$=\frac{65}{13}$