When Is the Projection of a Vector Undefined or Zero?

[emma3001]

I was wondering if it is possible, in projections, to have a projected onto b equal to zero or undefined. In other words, when does a projected onto b equal zero and when is it undefined?

 

[HallsofIvy]

It is certainly possible, in fact common, to have the projection of vector a onto vector b equal to 0: as long as a is perpendicular to b.

The "projection of a onto b" would be "undefined" if b itself has length 0.

In general, the (scalar) projection of a on b is
[tex]\frac{\vec{a}\cdot\vec{b}}{||b||}[/itex]
That's 0 if the dot product of a and b is 0 (a is perpendicular to b) and undefined if ||b||= 0.