LinAlg4.2.11

From Exampleproblems

Find a vector of magnitude 3 and which is perpendicular to both the vectors $3\bar{i}+\bar{j}-4\bar{k},6\bar{i}+5\bar{j}-2\bar{k}\,$

Let a and b be the two given vectors $3\bar{i}+\bar{j}-4\bar{k},6\bar{i}+5\bar{j}-2\bar{k}\,$

Then $\bar{a}\times\bar{b}=\begin{vmatrix} \bar{i} & \bar{j} & \bar{k} \\ 3 & 1 & -4 \\ 6 & 5 & -2 \end{vmatrix}\,$ which is equal to

$\bar{i}(-2+20)-\bar{j}(-6+24)+\bar{k}(15-6)=18\bar{i}-18\bar{j}+9\bar{k}\,$

Unit vector perpendicular to a and b is $\pm\frac{\bar{a}\times\bar{b}}{|a\times b|}=\pm \frac{9(2\bar{i}-2\bar{j}+\bar{k}}{9\sqrt{4+4+1}}=\pm\frac{1}{3}(2\bar{i}-2\bar{j}+\bar{k})\,$

A vector of magnitude 3 and perpendicular to both the given vectors is $\frac{3(2\bar{i}-2\bar{j}+\bar{k})}{3}=\pm (2\bar{i}-2\bar{j}+\bar{k})\,$

Main Page:Linear Algebra

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