Understanding Perpendicular Vector Components

[fightboy]

"Is it possible for a vector that has nonzero magnitude to have a component in some direction that is equal to zero?"
The answer key said that any vector that has a nonzero magnitude will always have a component of zero length in the direction perpendicular to the vector.

I'm having trouble visualizing this. Why will the vector always have a component of zero length?
If anyone could break this down for me it would be much appreciated!

 

[gneill]

Imagine a vector rising perpendicularly from a line (any given line). What's the projection of that vector onto the line?

An obvious analogy is the x-y axes of the Cartesian plane. How much "y" lies along the direction of "x"?

 

[rude man]

Draw a vector A on paper. Now, draw a vector B perpendicular to the A vector, both A and B have their startying ends together. Now draw a vector C from the tip of A to the tip of B. You obviously have C = A + B. How small does B have to be so that A = C?