how to determine vectors u and v are parallel, orthogonal, or neither?

Answer 1

If you know dot and cross products
if parallel, |A cross B| = |A| |B| sin theta
if parallel theta and sin theta = 0 and cross product is 0
if perpendicular then dot product = |A| |B| cos theta = 0

Answer 2

If you are not there yet
then
do y = m x + b for both
if the two m s are the same, same slope, parallel
if m for second = -1/m for first, then perpendicular