In 3D space, how to project a point into x, y and z?

Question

Let's say we have a point in a 3D space and we need to project the x component onto the x, y and z components onto the x-axis, y-axis and z-axis respectively with respect to the eigenvectors. How can we do that? The input: a point in 3D. the Origin <0,0,0> an eigenvectors $e_x, e_y$ and $e_z$ the output: a projected point onto x, y, and y-axis. does multiply the point by the eigenvectors (basis matrix) will solve this problem?

Concrete example: Here is a point P and the bisis vectors $ P=<0.348499, 2.137199, -0.438428> e_x = <0.944485, -0.244727, 0.219221> e_y = <0.275648, 0.953309, -0.12337> e_z = <-0.178793, 0.176948, 0.967844> $

Answer 1

You're actually already done if you're just projecting onto the standard coordinate axes:

$$P_X = <x,0,0>$$

$$P_Y = <0,y,0>$$

$$P_Z = <0,0,z>$$

Just take the point's coordinates and put them into three new points, with zeroes elsewhere.

However, if you are projecting onto an arbitrary unit vector, $e_i$, then the answer is to project the position vector of the point onto $\hat{e}_i$ as follows:

$$P_e = (P\cdot\hat{e}_i)\hat{e}_i$$

Where I've abused notation slightly an treated $P$ as its own position vector, which I'll do unashamedly.

Answer 2

The orthogonal projection of a vector $\mathbf v$ onto any other nonzero vector $\mathbf w$ is ${\mathbf v\cdot\mathbf w\over\mathbf w\cdot\mathbf w}\mathbf w$. If $\mathbf w$ happens to be a unit vector, then of course the denominator is equal to $1$.

The coordinates of a point are the ratios of the lengths of its projections onto the basis vectors and the lengths of the basis vectors. Those projections might not be orthogonal, but if the coordinate system is orthonormal then the point’s coordinates in that system are exactly the lengths of its orthogonal projections onto the basis vectors.