Module 2: The Cross Product

A second useful product computation is the cross product. But this operation is restricted only to vectors in LaTeX: \mathbb{R} ^{3} R3 (whereas the dot product is defined for vectors in any (natural) dimension.)

LaTeX: \mbox{Let } \vec{a} = \begin{pmatrix}
a_x \\ a_y \\ a_z
\end{pmatrix}
\mbox{ and }
\vec{b} = \begin{pmatrix}
b_x \\ b_y \\ b_z
\end{pmatrix}Let a=(axayaz) and b=(bxbybz)

Then we can define:

LaTeX: \vec{a} \times \vec{b} = \begin{pmatrix}
a_yb_z - a_zb_y \\
a_zb_x - a_xb_z \\
a_xb_y -a_yb_x
\end{pmatrix}.a×b=(aybzazbyazbxaxbzaxbyaybx).

This is a very awkward definition, but can be made nicer if we think about the pattern illustrated in the following picture:

mod23Artboard 2.png

For those of you who have computed the determinant of a three by three matrix, this pattern should be familiar.

Example:

LaTeX: \begin{pmatrix}
  -1 \\ 1 \\ 2
  \end{pmatrix}
  \times
  \begin{pmatrix}
  0 \\ 3 \\ -2
  \end{pmatrix}
  =
  \begin{pmatrix}
  -2-6 \\ 0-2 \\ -3+0
  \end{pmatrix}
  =
  \begin{pmatrix}
  -8 \\ -2 \\ -3
  \end{pmatrix}(112)×(032)=(26023+0)=(823)