LinAlg4.2.9

From Exampleproblems

Determine the unit vector perpendicular to both the vectors $2\bar{i}+\bar{j}+3\bar{k},\bar{i}-2\bar{j}+\bar{k}\,$

Let a,b be the given vectors $2\bar{i}+\bar{j}+3\bar{k},\bar{i}-2\bar{j}+\bar{k}\,$

$\bar{a}\times\bar{b}=\begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ 2 & 1 & 3 \\ 1 & -2 & 1 \end{vmatrix}\,$

which is equal to $\bar{i}(1+6)-\bar{j}(2-3)+\bar{k}(-4-1)=7\bar{i}+\bar{j}-5\bar{k}\,$

Therefore unit vector perpendicular to both the vectors is $\pm\frac{\bar{a}\times\bar{b}}{|a||b|}=\pm\frac{7\bar{i}+\bar{j}-5\bar{k}}{\sqrt{49+1+25}}=\pm\frac{1}{5\sqrt{3}}(7\bar{i}+\bar{j}-5\bar{k})\,$

Main Page:Linear Algebra

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