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So, i wanna find points on a plane that are equidistants from :

A=(1,1,0) B=(0,1,1)

π : y = x

what i tried: Set x = 0 -> y=0, z=0.

N(pi) = (1,-1,0)
P1 = (0,0,0);

Dist from pi to A = ((1*1-1*1+0*0) / sqrt(1+1+0) ) = 0;

so i try to figure out what is the distance from pi to B.

Dist from pi to B = ((1*0+(-1)*(-1)+0*1)/sqrt(1+1+0)) = 1/sqrt(2).

but i dont know how to discover the points that are equidistants from A and B from pi.

Thanks.

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up vote 1 down vote accepted

So, you can conclude at least that $A\in\pi$. Now if a point $X$ is equidistant from $A$ and the plane $\pi$, then $AX$ has to be the distance segment itself, hence orthogonal to $\pi$, i.e. $X=A+\lambda\,(1,-1,0)$. (Its distance from $A$ is $|\lambda|\sqrt2$.)

Now $\vec{BX}=A+\lambda\,(1,-1,0)-B=(1+\lambda, -\lambda,-1)$. Its length is $\sqrt{(1+\lambda)^2+\lambda^2+1}$, so we need $$ \begin{align} (1+\lambda)^2+\lambda^2+1 &=2\lambda^2 \\ 2+2\lambda &=0 \\ \lambda &=-1\,. \end{align}$$

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the distance from A is λ√2, based on 1/√2? i dont understand this part only. –  Matheus Silva Nov 17 '13 at 1:03
    
$\vec{AX}=X-A=\lambda\cdot(1,-1,0)$. And the length of the vector $(1,-1,0)$ is $\sqrt{1+1+0}=\sqrt2$. –  Berci Nov 17 '13 at 2:22
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