Editing Cross product (section)

=== Cross visualization ===
Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula.

If

:<math>\mathbf{a} = \mathbf{b} \times \mathbf{c}</math>

then:

:<math>
  \mathbf{a} =
    \begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix} \times
    \begin{bmatrix}c_x\\c_y\\c_z\end{bmatrix}.
</math>

If we want to obtain the formula for <math>a_x</math> we simply drop the <math>b_x</math> and <math>c_x</math> from the formula, and take the next two components down:

:<math>
  a_x =
    \begin{bmatrix}b_y\\b_z\end{bmatrix} \times
    \begin{bmatrix}c_y\\c_z\end{bmatrix}.
</math>

When doing this for <math>a_y</math> the next two elements down should "wrap around" the matrix so that after the z component comes the x component. For clarity, when performing this operation for <math>a_y</math>, the next two components should be z and x (in that order). While for <math>a_z</math> the next two components should be taken as x and y.

:<math>
  a_y =
    \begin{bmatrix}b_z\\b_x\end{bmatrix} \times
    \begin{bmatrix}c_z\\c_x\end{bmatrix},\ 
  a_z =
    \begin{bmatrix}b_x\\b_y\end{bmatrix} \times
    \begin{bmatrix}c_x\\c_y\end{bmatrix}
</math>

For <math>a_x</math> then, if we visualize the cross operator as pointing from an element on the left to an element on the right, we can take the first element on the left and simply multiply by the element that the cross points to in the right-hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our <math>a_x</math> formula –

:<math>a_x = b_y c_z - b_z c_y.</math>

We can do this in the same way for <math>a_y</math> and <math>a_z</math> to construct their associated formulas.