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Suppose I have a point $(a,b,c)$. What is the distance from this point to the $x$-axis?

I had supposed that it would be simply $\sqrt{b^2+c^2}$ but this does not seem to be the case.

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    I had the very same intuition- turns out I added the $y^2$ and $z^2$ in question incorrectly.2012-08-29

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In fact is $\sqrt{b^2+c^2}$. You can see this if you think about the point $(a,b,c)$ in the plane $(a,0,0)$.