LinAlg4.2.9

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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}})\,


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