Question

Can a vector have zero component along a line and still have nonzero magnitude?

Answer 1

Yes, a vector can have zero components along a line and still have a nonzero magnitude.
Example: Consider a two dimensional vector $2\hat{i}+0\hat{j}$. This vector has zero components along a line lying along the Y-axis and a nonzero component along the X-axis. The magnitude of the vector is also nonzero.
Now, magnitude of $2\hat{i}+0\hat{j}$ = $\sqrt{2^2+0^2}=2$