Projection in Plane on X-Axis along Y-Axis

Theorem
Let $\pr_{X, Y}$ denote the projection on the $x$-axis along the $y$-axis:
 * $\forall P \in \R^2: \map {\pr_{X, Y} } P =$ the intersection of the $x$-axis with the line through $P$ parallel to the $y$-axis.

Let $P = \tuple {\lambda_1, \lambda_2}$ be an arbitrary point in $\R^2$.

Then:
 * $\map {\pr_{X, Y} } {\lambda_1, \lambda_2} = \tuple {\lambda_1, 0}$

Proof
This is an instance of a projection on $M$ along $N$ where $N$ coincides with the $y$-axis.

Hence it is one of the special cases of Equations defining Projection in Plane: Cartesian:


 * $\map {\pr_{M, N} } P = \tuple {\lambda_1, \lambda_1 \tan \theta}$

where $\theta$ is the angle between $M$ and the $x$-axis.

In this case, $M$ is actually coincident with the $x$-axis.

Hence $\theta = 0$ and so from Tangent of Zero:


 * $\map {\pr_{X, Y} } P = \tuple {\lambda_1, 0}$