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I have a point $P = (0, y, z)$. I need to determine the $x$ coordinate of its projection $P'$ onto a plane $T$ along the $x$ axis unit vector $\vec{X} = (1, 0, 0)$. The plane $T$ is defined by a point $V$ that lies on $T$ and its normal $\vec{N}$ (see the attached picture).

Projection onto plane along a vector.

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You can determine x by calculating line-plane intersection. Your line starts at P and has direction D=[1, 0, 0], and plane T=[V, N]. This can be done as follows: $$x=\frac{(V-P)\cdot N}{N\cdot D}$$ x is the distance from P along D to the intersection point with T (assuming both D and N are unit vectors)

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