# What is the cross product of (1,3,4) xx (-1,0,-1)?

May 11, 2018

$\left(1 , 3 , 4\right) \times \left(- 1 , 0 , - 1\right) = \left(- 3 , - 3 , 3\right)$

#### Explanation:

$\left(1 , 3 , 4\right) \times \left(- 1 , 0 , - 1\right)$

Can be written as a determinant:

(1,3,4) xx (-1,0,-1) =| (hati,hatj,hatk), (1,3,4), (-1,0,-1) |

Subtract the product of the minor diagonals from the product of the major diagonals:

$\left(1 , 3 , 4\right) \times \left(- 1 , 0 , - 1\right) = \left\{\left(3\right) \left(- 1\right) - \left(4\right) \left(0\right)\right\} \hat{i} + \left\{\left(4\right) \left(- 1\right) - \left(1\right) \left(- 1\right)\right\} \hat{j} + \left\{\left(1\right) \left(0\right) - \left(3\right) \left(- 1\right)\right\} \hat{k}$

$\left(1 , 3 , 4\right) \times \left(- 1 , 0 , - 1\right) = - 3 \hat{i} - 3 \hat{j} + 3 \hat{k}$

Change notation:

$\left(1 , 3 , 4\right) \times \left(- 1 , 0 , - 1\right) = \left(- 3 , - 3 , 3\right)$