Given two nonzero vectors
and
, we have
where is the angle between the two vectors, and
and
are respectively the magnitudes of vectors
and
.. Note that
since both vectors are non-zero.
Let's assume that both the dot and cross product of two non-zero vectors can be zero. Then
This further yieds
Hence, if both the dot and cross product of two non-zero vectors were zero, then the angle between the two vectors should satisfy the following condition
However, this condition cannot be satisfied for any angle . For example, if then , but . On the other hand, if then , but .
We have reached a contradiction.
Conclusion: for two non-zero vectors,
and
, both their dot and cross product cannot be zero!