Problem 89
Easy Vector Product

Compute |vec a×2vec b| if |vec a|=6 , |vec b|=5 , angle(vec…

Compute $|\vec a\times2\vec b|$ if $|\vec a|=6$, $|\vec b|=5$, $\angle(\vec a,\vec b)=\dfrac\pi6$.

Explanation of Method

The magnitude of a×2b.

Step 1: Computation

\[ |\vec a\times2\vec b|=2|\vec a\times\vec b|=2\cdot6\cdot5\cdot\sin\frac\pi6=2\cdot15. \]

\[|\vec a\times2\vec b|=30\]

$\tfrac{\pi}{6}$ $O$ $\vec a$ $2\vec b$ $h=10\sin\tfrac\pi6=5$ $S=30$
$|\vec a\times2\vec b|$ is the area of the parallelogram on $\vec a$ ($6$) and $2\vec b$ ($10$): base $6$ times height $5$.

Result

\[\boxed{\,|\vec a\times2\vec b|=30\,}\]