About the orthogonal projection.

Question

The projection

$$Proj_{\vec{AC}}\vec{AB}$$

Yields the vector that corresponds to the line.

I am unsure, however:

• Is it necessary for $\vec{AB}$ and $\vec{AC}$ to be adjacent? All the examples I see, they are, but dunno.
• Does the length of $\vec{AC}$ have any sort of impact in the result? From what I gather, $\vec{AC}$ is used merely to describe the angle so that the projection can be calculated.
• When it comes to orthogonal projection, the examples always use localised arrows ($\vec{AB}$ and $\vec{AC}$) - must that be the case? What happens if you do it with two points $X$ and $Y$ in space instead? (I don't see any reason to do so, but just curious).

Answer 1

• Projection is defined on vectors. Vectors have a magnitude and direction (think of them like displacements); the position of the base of the arrow has no meaning. Thus, it's meaningless to say that $\vec{AB}$ and $\vec{AC}$ are adjacent.
• The length of $\vec{AC}$ has no impact on the final vector from the projection.
• Again, projection is defined on vectors, what you call "localized arrows". It doesn't mean anything to project one position on another.