Cross Product

## Description

The cross product of any two vectors $a = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ and $a = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$ in $\mathbb{R} ^ 3$ is is defined as

(1)\begin{align} \mathbf a \times \mathbf b = \begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1-a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{pmatrix}. \end{align}

The Sage cell below calculates the dot product for the vectors $a = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $b = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$.

## Sage Cell

#### Code

```
a = vector( [ 1, 2, 3 ] )
b = vector( [ 4, 5, 6 ] )
crossproduct=a.cross_product(b)
crossproduct
```

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Date: 06 Nov 2018 16:40

Submitted by: James A Phillips