If $|\vec a|=5,|\vec b|=3,|\vec c|=4$ and $\vec a$ is perpendicular to $\vec b$ and $\vec c$ such that angle between $\vec b$ and $\vec c$ is $\frac{5π}{6}$, then the volume of the parallelopiped having $\vec a, \vec b$ and $\vec c$ as three coterminous edges is
Answer & explanation
Correct answer: option 1
Since $\vec a$ is perpendicular to both, $\vec b$ and $\vec c$.
Therefore, $\vec a$ is parallel to $\vec b×\vec c$.
Now,
$\left|[\vec a\,\,\vec b\,\,\vec c]\right|=\left|\vec a.(\vec b×\vec c)\right|$
$⇒\left|[\vec a\,\,\vec b\,\,\vec c]\right|=|\vec a||\vec b×\vec c|$ $[∵\vec a||\vec b×\vec c]$
$⇒\left|[\vec a\,\,\vec b\,\,\vec c]\right|=|\vec a||\vec b||\vec c|\sin\frac{5π}{6}$
$⇒\left|[\vec a\,\,\vec b\,\,\vec c]\right|=5×3×4×\frac{1}{2}=30$ cubic units