LinAlg4.2.15

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Find the volume of the parallelopiped whose edges are {\bar  {i}}+{\bar  {j}}+{\bar  {k}},{\bar  {i}}-{\bar  {j}}+{\bar  {k}},{\bar  {i}}+{\bar  {j}}-{\bar  {k}}\,.

Given edges of the parallelopiped are {\bar  {i}}+{\bar  {j}}+{\bar  {k}},{\bar  {i}}-{\bar  {j}}+{\bar  {k}},{\bar  {i}}+{\bar  {j}}-{\bar  {k}}\,.

Volume of the parallelopiped is |[{\bar  {i}}+{\bar  {j}}+{\bar  {k}},{\bar  {i}}-{\bar  {j}}+{\bar  {k}},{\bar  {i}}+{\bar  {j}}-{\bar  {k}}]|\,. which is equal to

{\begin{vmatrix}1&1&1\\1&-1&1\\1&1&-1\end{vmatrix}}\,

=|1(1-1)-1(-1-1)+1(1+1)=|2+2|=4\, cubic units.

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