LinAlg4.2.11

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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}})\,


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