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Section 11.4 : Cross Product

4. Are the vectors \(\vec u = \left\langle {1,2, - 4} \right\rangle \), \(\vec v = \left\langle { - 5,3, - 7} \right\rangle \) and \(\vec w = \left\langle { - 1,4,2} \right\rangle \) are in the same plane?

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As discussed in the notes to answer this question all we need to do is compute the following quantity,

\[\begin{align*}\vec u\centerdot \left( {\vec v \times \vec w} \right) & = \left| {\begin{array}{*{20}{c}}1&2&{ - 4}\\{ - 5}&3&{ - 7}\\{ - 1}&4&2\end{array}} \right|\,\,\,\,\,\,\,\begin{array}{*{20}{c}}1&2\\{ - 5}&3\\{ - 1}&4\end{array}\\ & = 6 + 14 + 80 - \left( { - 20} \right) - \left( { - 28} \right) - 12 = 136\end{align*}\]

Okay, since this is not zero we know that they are not in the same plane.