How can I find the z Componet of a 3d line when I have the x and y of a point on that line and the lines direction?

Question

So Basically I have the x and y of a 3D point and I need to find the z. I have access to other points on this 3D line to determine direction / slope. No clue where to begin.

Answer 1

Assuming you have two different points $u_i = (x_i, y_i, z_i)$, for $i \in \{ 1, 2 \}$ from the line. Then you can describe all points on the line as $$ u = (1 - \lambda) u_1 + \lambda u_2 \quad (\lambda \in \mathbb{R}) \quad (*) $$ You can then try to find parameter values for a given point $(x,y,z)$, where only $x$ and $y$ are known: $$ x = (1 - \lambda) x_1 + \lambda x_2 = x_1 + \lambda (x_2 - x_1) \iff \\ x - x_1 = \lambda (x_2 - x_1) $$ To solve for $\lambda$, we need $x_1 \ne x_2$.

Then $$ \lambda = \frac{x-x_1}{x_2-x_1} $$ and we can check the point $u$ given by equation $(*)$ if its first two coordinates agree with the given $x$ and $y$ and if they do, assume the third coordinate as sought $z$ value.

If $x_1 = x_2$ we need to resort to $$ y - y_1 = \lambda(y_2 -y_1) $$ and so on. If $y_1 \ne y_2$ then we do the same as above, but for the $y$-coordinate.

This leaves the case $x_1 = x_2$ and $y_1 = y_2$. We asked for different points so we have $z_1 \ne z_2$. The line runs parallel to the $z$-axis.

For this case all points on the line match your given $x$ and $y$ coordinates, you have no way to determine which one is yours, without additional information.

Answer 2

Let two other points be $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$. Then, the equation of the line is given by: $$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$$ Substitute the known values of $x$ and $y$ in this to get the value of $z$.