# What is the cross product of [0,8,5] and [1,2,-4] ?

Apr 1, 2016

$\left[0 , 8 , 5\right] \times \left[1 , 2 , - 4\right] = \left[- 42 , 5 , - 8\right]$

#### Explanation:

The cross product of $\vec{A}$ and $\vec{B}$ is given by

$\vec{A} \times \vec{B} = | | \vec{A} | | \cdot | | \vec{B} | | \cdot \sin \left(\theta\right) \hat{n}$,

where $\theta$ is the positive angle between $\vec{A}$ and $\vec{B}$, and $\hat{n}$ is a unit vector with direction given by the right hand rule.

For the unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$ in the directions of $x$, $y$ and $z$ respectively,

$\textcolor{w h i t e}{\begin{matrix}\textcolor{b l a c k}{\hat{i} \times \hat{i} = \vec{0}} & \textcolor{b l a c k}{q \quad \hat{i} \times \hat{j} = \hat{k}} & \textcolor{b l a c k}{q \quad \hat{i} \times \hat{k} = - \hat{j}} \\ \textcolor{b l a c k}{\hat{j} \times \hat{i} = - \hat{k}} & \textcolor{b l a c k}{q \quad \hat{j} \times \hat{j} = \vec{0}} & \textcolor{b l a c k}{q \quad \hat{j} \times \hat{k} = \hat{i}} \\ \textcolor{b l a c k}{\hat{k} \times \hat{i} = \hat{j}} & \textcolor{b l a c k}{q \quad \hat{k} \times \hat{j} = - \hat{i}} & \textcolor{b l a c k}{q \quad \hat{k} \times \hat{k} = \vec{0}}\end{matrix}}$

Also, cross product is distributive, which means

$\vec{A} \times \left(\vec{B} + \vec{C}\right) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$.

For this question,

$\left[0 , 8 , 5\right] \times \left[1 , 2 , - 4\right]$

$= \left(8 \hat{j} + 5 \hat{k}\right) \times \left(\hat{i} + 2 \hat{j} - 4 \hat{k}\right)$

$= \textcolor{w h i t e}{\begin{matrix}\textcolor{b l a c k}{q \quad 8 \hat{j} \times \hat{i} + 8 \hat{j} \times 2 \hat{j} + 8 \hat{j} \times \left(- 4 \hat{k}\right)} \\ \textcolor{b l a c k}{+ 5 \hat{k} \times \hat{i} + 5 \hat{k} \times 2 \hat{j} + 5 \hat{k} \times \left(- 4 \hat{k}\right)}\end{matrix}}$

$= \textcolor{w h i t e}{\begin{matrix}\textcolor{b l a c k}{- 8 \hat{k} + 16 \left(\vec{0}\right) - 32 \hat{i}} \\ \textcolor{b l a c k}{q \quad + 5 \hat{j} - \quad 10 \hat{i} \quad - 20 \left(\vec{0}\right)}\end{matrix}}$

$= - 42 \hat{i} + 5 \hat{j} - 8 \hat{k}$

$= \left[- 42 , 5 , - 8\right]$