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  4. If veca=2 i+j-3 k and vecb=i-2 j+k then a vector of magnitude 5 perpendicular to both veca and vecb is

Q. If $\vec{a}=2 i+j-3 k$ and $\vec{b}=i-2 j+k$ then a vector of magnitude $5$ perpendicular to both $\vec{a}$ and $\vec{b}$ is

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Solution:

$\vec{a} \times \vec{b}=\begin{vmatrix}i & j & k \\ 2 & 1 & -3 \\ 1 & -2 & 1\end{vmatrix}=-5 i-5 j-5 k $
$|\vec{a} \times \vec{b}|=\sqrt{25+25+25}=5 \sqrt{3}$
$\therefore$ Vector of magnitude $5$ perpendicular to both $\vec{a}$ and $\vec{b}$ is
$\pm 5 \hat{n}=\pm 5\left(\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}\right)=\pm \frac{5}{\sqrt{3}}(-i-j-k)$

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