Linda Peters

2022-09-27

Suppose $\stackrel{\to }{v}$ and $\stackrel{\to }{w}$ are two vectors parallel to the plane
$x+2y+3z=7.$
Suppose furthermore that $\stackrel{\to }{v}$ is perpendicular to $\stackrel{\to }{w}$ ,

How would you go about answering this question? I always reach a dead end when I try to solve $\stackrel{\to }{v}$ and $\stackrel{\to }{w}$ that satisfy the given information, I just don't know how to go about it. I tried to draw the plane and two parallel vectors, then I know the length of the cross product would be 12. Then what do I do now? How could you solve this to find vectors $\stackrel{\to }{v}$ and $\stackrel{\to }{w}$?

ahem37

You have:
$\stackrel{\to }{u}×\stackrel{\to }{v}=n\left(1,2,3\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|\stackrel{\to }{u}×\stackrel{\to }{v}|=|n|\sqrt{{1}^{2}+{2}^{2}+{3}^{2}}=|n|\sqrt{14}=12\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}n=±\frac{12}{\sqrt{14}}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\stackrel{\to }{u}×\stackrel{\to }{v}=±\frac{12}{\sqrt{14}}\left(1,2,3\right)$

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