What is the meaning of orthogonal projection of vectors?

Question

The vector $\dfrac{x\cdot y}{y\cdot y}y$ is called the projection of $x$ along $y$.

I do not understand what it means, geometrically. Can anyone give me n specific example of what it means?

Answer 1

In two dimensions, a projection onto a line is a transformation that moves every point in the plane onto the line, but in a way that keeps each point of the line unmoved.

Orthogonal projection means the specific projection that moves each point in a direction orthogonal to the line.

The plane and line are not special: the same idea can be used in $n$ dimensions to project onto an $m$-dimensional affine subspace.

Your formula is for the special case where you're projecting onto a line passing through the origin. $x$ is the vector describing the point you're projecting, and $y$ is any vector parallel to the line.

You can see this by observing that if $z$ is any vector orthogonal to $y$, then $x$ and $x+z$ both get mapped to the same place. And if you replace $y$ with any nonzero multiple $ry$, it doesn't change the values you get from the formula.