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