(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A plane has the equation aX + bY + cZ + d = 0. A line L goes from (0,0,0) and crosses the plane at some point. L and plane are orthogonal. express the coordinates of the crossing point P by: a, b, c and d.

2. Relevant equations

|d|/sqrt(a^2 + b^2 + c^2) is the distance between point P and origo.

[a,b,c] is the directional vector for the line and a normal vector on the plane.

3. The attempt at a solution

So, if I divide [a,b,c] by the length of the directional vector, sqrt(a^2 + b^2 + c^2) and then multiply it by the length of the OP vector, |d|/sqrt(a^2 + b^2 + c^2), I get:

a|d|/(a^2 + b^2 + c^2), b|d|/(a^2 + b^2 + c^2), c|d|/(a^2 + b^2 + c^2)=P.

This is the crossing point of the line and the plane (or a point completely opposite to the plane).

The answer is supposed to be (-ad/(a^{2}+ b^{2}+ c^^{2}),-bd/(a^{2}+ b^{2}+ c^{2}),-cd/(a^{2}+ b^{2}+ c^{2}))=P

I know that my answer is completely wrong, because |d| will remain always positive, while [a,b,c] could be negative or positive, with no consequence to |d|.

Can somebody tell me how I can get to the correct answer?

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# Wrong answer using correct logic

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