# What is the cross product of [9,4,-1] and [2, 5, 4] ?

Dec 13, 2015

The cross product of two 3D vectors is another 3D vector orthogonal to both.

The cross product is defined as:

$\textcolor{g r e e n}{\vec{u} \times \vec{v} = \left\langle{u}_{2} {v}_{3} - {u}_{3} {v}_{2} , {u}_{3} {v}_{1} - {u}_{1} {v}_{3} , {u}_{1} {v}_{2} - {u}_{2} {v}_{1}\right\rangle}$

It is easier to remember it if we remember that it starts with $2 , 3 - 3 , 2$, and is cyclic and antisymmetric.

• it cycles as $2 , 3$ $\to$ $3 , 1$ $\to$ $1 , 2$
• it is antisymmetric in that it goes: $2 , 3$ // $3 , 2$ $\to$ $3 , 1$ // $1 , 3$ $\to$ $1 , 2$ // $2 , 1$, but subtracts each pair of products.

So, let:

$\vec{u} = \left\langle9 , 4 , - 1\right\rangle$
$\vec{v} = \left\langle2 , 5 , 4\right\rangle$

$\vec{u} \times \vec{v}$

$= \left\langle\left(4 \times 4\right) - \left(- 1 \times 5\right) , \left(- 1 \times 2\right) - \left(9 \times 4\right) , \left(9 \times 5\right) - \left(4 \times 2\right)\right\rangle$

$= \left\langle16 - \left(- 5\right) , - 2 - 36 , 45 - 8\right\rangle$

$= \textcolor{b l u e}{\left\langle21 , - 38 , 37\right\rangle}$