Paul's Online Notes
Home / Calculus II / Vectors / Cross Product
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### 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?

Show Solution

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.