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Math Help - Distance between dot and surface

  1. #1
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    Distance between dot and surface

    Find the distance between (1,0,0,-2) in Euclidean space R^4 and a surface L: \left\{\begin{array}{cc}x+y+z-w=0&\mbox{}<br />
\\2x-y+w=0 & \mbox{}\end{array}\right.
    Thanks to anyone able to solve it.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by wicked View Post
    Find the distance between (1,0,0,-2) in Euclidean space R^4 and a surface L: \left\{\begin{array}{cc}x+y+z-w=0&\mbox{}<br />
\\2x-y+w=0 & \mbox{}\end{array}\right.
    Thanks to anyone able to solve it.
    Your "surface" is a (two-dimensional) vectorial plane. Find an orthonormal basis (u,v) of L (i.e. find two non-colinear vectors in L and orthonormalize them), then the projection of x=(1,0,0,-2) on L is \pi_L(x)=(x,u)u+(x,v)v (dot products) and thus by "Pythagoras theorem" {\rm dist}(x,L)^2=\|x\|^2-\|\pi_L(x)\|^2=\|x\|^2-(x,u)^2-(x,v)^2. I find 3/2 but I may very well be wrong.
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