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My textbook makes a quick, unproven claim that if a vector P is orthogonal to /one/ vector that lies on a plane G, then P is normal to the plane. My question is, is this a mistake? Wouldn't you need P to be orthogonal to two vectors on G not contained in the same line, for it to be normal to G?

Thanks!

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It's clearly a mistake. The line $y=z,x=0$ is orthogonal to the $x$-axis but not to the $xy$-plane. –  Brian M. Scott Apr 30 '12 at 20:55
    
Thanks Brian. So is my condition correct? That is, for a vector to be normal to a plane it needs to be orthogonal to two vectors on it that aren't on the same line? –  r34 Apr 30 '12 at 20:57
    
Yes, because if it's normal to $u$ and $v$, then it's normal to everything in the span of $u$ and $v$. –  Brian M. Scott Apr 30 '12 at 20:59
    
Thank you for your help. You can post this as an answer if you like, so we can close this question. –  r34 Apr 30 '12 at 21:00
    
Done!$\qquad\qquad$ –  Brian M. Scott Apr 30 '12 at 21:02

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up vote 1 down vote accepted

You're quite right: it's clearly a mistake, since the line $y=z,x=0$ is orthogonal to the $x$-axis but not to the $xy$-plane. Your suggested revision is correct, because if a vector is normal to $u$ and $v$, it's normal to everything in the span of $u$ and $v$.

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