Editing Cross product (section)

=== Mnemonic ===
[[File:cross_product_mnemonic.svg|thumb|Mnemonic to calculate a cross product in vector form]]
{{redirect|Xyzzy (mnemonic)||Xyzzy (disambiguation){{!}}Xyzzy}}

The word "xyzzy" can be used to remember the definition of the cross product.

If

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

where:

:<math>
  \mathbf{a} = \begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix},\ 
  \mathbf{b} = \begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix},\ 
  \mathbf{c} = \begin{bmatrix}c_x\\c_y\\c_z\end{bmatrix}
</math>

then:

:<math>a_x = b_y c_z - b_z c_y </math>
:<math>a_y = b_z c_x - b_x c_z </math>
:<math>a_z = b_x c_y - b_y c_x. </math>

The second and third equations can be obtained from the first by simply vertically rotating the subscripts, {{nowrap|''x'' → ''y'' → ''z'' → ''x''}}. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus's scheme (those containing '''''i'''''), or to remember the xyzzy sequence.

Since the first diagonal in Sarrus's scheme is just the [[main diagonal]] of the [[#Matrix notation|above]]-mentioned 3×3 matrix, the first three letters of the word xyzzy can be very easily remembered.