Vector Cross Product Table

This table shows the resulting force vector (\(\vec{F}\)) from the cross product of velocity (\(\vec{v}\)) and magnetic field (\(\vec{B}\)).

Velocity (\(\vec{v}\))	Magnetic Field (\(\vec{B}\))	Resulting Force (\(\vec{F} = \vec{v} \times \vec{B}\))
\(\hat{i}\) (East)	\(\hat{i}\) (East)	0
\(\hat{i}\) (East)	\(\hat{j}\) (North)	\(\hat{k}\)
\(\hat{i}\) (East)	\(\hat{k}\) (Up)	\(-\hat{j}\)
\(\hat{i}\) (East)	\(-\hat{i}\) (West)	0
\(\hat{i}\) (East)	\(-\hat{j}\) (South)	\(-\hat{k}\)
\(\hat{i}\) (East)	\(-\hat{k}\) (Down)	\(\hat{j}\)
\(\hat{j}\) (North)	\(\hat{i}\) (East)	\(-\hat{k}\)
\(\hat{j}\) (North)	\(\hat{j}\) (North)	0
\(\hat{j}\) (North)	\(\hat{k}\) (Up)	\(\hat{i}\)
\(\hat{j}\) (North)	\(-\hat{i}\) (West)	\(\hat{k}\)
\(\hat{j}\) (North)	\(-\hat{j}\) (South)	0
\(\hat{j}\) (North)	\(-\hat{k}\) (Down)	\(-\hat{i}\)
\(\hat{k}\) (Up)	\(\hat{i}\) (East)	\(\hat{j}\)
\(\hat{k}\) (Up)	\(\hat{j}\) (North)	\(-\hat{i}\)
\(\hat{k}\) (Up)	\(\hat{k}\) (Up)	0
\(\hat{k}\) (Up)	\(-\hat{i}\) (West)	\(-\hat{j}\)
\(\hat{k}\) (Up)	\(-\hat{j}\) (South)	\(\hat{i}\)
\(\hat{k}\) (Up)	\(-\hat{k}\) (Down)	0
\(-\hat{i}\) (West)	\(\hat{i}\) (East)	0
\(-\hat{i}\) (West)	\(\hat{j}\) (North)	\(-\hat{k}\)
\(-\hat{i}\) (West)	\(\hat{k}\) (Up)	\(\hat{j}\)
\(-\hat{i}\) (West)	\(-\hat{i}\) (West)	0
\(-\hat{i}\) (West)	\(-\hat{j}\) (South)	\(\hat{k}\)
\(-\hat{i}\) (West)	\(-\hat{k}\) (Down)	\(-\hat{j}\)
\(-\hat{j}\) (South)	\(\hat{i}\) (East)	\(\hat{k}\)
\(-\hat{j}\) (South)	\(\hat{j}\) (North)	0
\(-\hat{j}\) (South)	\(\hat{k}\) (Up)	\(-\hat{i}\)
\(-\hat{j}\) (South)	\(-\hat{i}\) (West)	\(-\hat{k}\)
\(-\hat{j}\) (South)	\(-\hat{j}\) (South)	0
\(-\hat{j}\) (South)	\(-\hat{k}\) (Down)	\(\hat{i}\)
\(-\hat{k}\) (Down)	\(\hat{i}\) (East)	\(-\hat{j}\)
\(-\hat{k}\) (Down)	\(\hat{j}\) (North)	\(\hat{i}\)
\(-\hat{k}\) (Down)	\(\hat{k}\) (Up)	0
\(-\hat{k}\) (Down)	\(-\hat{i}\) (West)	\(\hat{j}\)
\(-\hat{k}\) (Down)	\(-\hat{j}\) (South)	\(-\hat{i}\)
\(-\hat{k}\) (Down)	\(-\hat{k}\) (Down)	0