Checking Parallelism Using Cross Products

There are a number of ways to check if two vectors $\vec{a}$ and $\vec{b}$ are parallel.

Two vectors are parallel if and only if their cross product is zero. Be warned that the cross product is only defined for three-dimensional vectors, although 2-D vectors may be easily transformed into 3-D vectors by adding a third component of zero.

Two vectors $\vec{a}$ and $\vec{b}$ are parallel if and only if there exists some real number $k$ such that $k\vec{a}=\vec{b}$. This is by far the most common and easiest test of parallelism.

Third, two vectors are parallel if and only if their dot product is equal to the product of their magnitudes, up to sign.

Note that all of these criteria lead us to the conclusion that the zero vector is parallel to all vectors; this result is peculiar since the zero vector has no preferred direction. This oddity is usually accepted, since it is not very meaningful anyway. In rare cases the zero vector is discarded in discussions of parallelism, in which case the scale factor $k$ is restricted to be nonzero.