In proving the formula of the distance between a plane and a point, why is the vector joining the point and an arbitrary point in the plane projected onto the unit normal vector? If it projected onto the normal vector itself, will make the formula wrong?
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Are you referring to finding the distance from the point to the plane using an arbitrary point in the plane? – atul ganju Mar 13 '22 at 21:14
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@justaguy, yes, – wawar05 Mar 13 '22 at 21:30
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Take some point $x$, and a plane $H = \{x : \langle a, x \rangle = b\}$. If we define the function $f$ to be return the vector that goes from $x$ to the closest point to it in $H$, we have: $$f(x,H) = \frac{\langle a, x-y \rangle}{\|a\|^2}a$$ where we can take $y$ to be any point on $H$. You can then solve for the distance from the point to the plane by taking the magnitude of the vector outputted by $H$. Notice, if $a$ is given as a unit vector then the formula can be written as: $$f(x,H) = \langle a, x-y \rangle$$
atul ganju
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