LinAlg4.2.10

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Find the vector area of a parallelogram whose diagonals are determined by the vectors {\bar  {a}}=3{\bar  {i}}+{\bar  {j}}-2{\bar  {k}},{\bar  {b}}={\bar  {i}}-3{\bar  {j}}+4{\bar  {k}}\,

{\bar  {a}}\times {\bar  {b}}={\begin{vmatrix}{\bar  {i}}&{\bar  {j}}&{\bar  {k}}\\3&1&-2\\1&-3&4\end{vmatrix}}\,

{\bar  {i}}(4-6)-{\bar  {j}}(2+2)+{\bar  {k}}(-9-1)=-2{\bar  {i}}-14{\bar  {j}}-10{\bar  {k}}\,

Therefore,area of the parallelogram = {\frac  {1}{2}}|a\times b|=|{\frac  {1}{2}}{\sqrt  {(-2)^{2}+(-14)^{2}+(-10)^{2}}}|={\frac  {1}{2}}{\sqrt  {300}}=5{\sqrt  {3}}\, units.


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