### Cross Product

• Evaluate cross product.

• Understand physical interpretation of cross product and its applications.

• Understand properties and identities related to cross product.

### Solved Example:

#### 4-7-01

Cross product of two unit vectors has a magnitude = 1. The $\angle$ between the vectors will be:

Solution:
$\left | \vec{a} \times \vec{b}\right | = a b \sin \theta$ $1 = (1) (1) \sin \theta$ $\theta = 90^\circ$

### Solved Example:

#### 4-7-02

What is the cross product $\overline{A}$ x $\overline{B}$ of the vectors, $\overline{A}$ = $\overline{i}$ + 4$\overline{j}$ + 6$\overline{k}$ and $\overline{B}$ = 2$\overline{i}$ + 3$\overline{j}$ + 5$\overline{k}$ ?

Solution:
$\overline {A}=\overline{i}+4\overline {j}+6\overline {k}$ $\overline {B}=2\overline {i}+3\overline {j}+5\overline {k}$ \begin{align*} \overline {A} \times \overline {B} &= \begin{vmatrix} \overline{i} & \overline{j} & \overline{k}\\ 1 & 4 & 6 \\ 2& 3 & 5 \end{vmatrix}\\ &= \overline {i}\left( 20-18\right) -\overline {j}\left( 5-12\right) +\overline {k}\left( 3-8\right)\\ &=2\overline {i}+7\overline {j}-5\overline {k} \end{align*}

Find the magnitude of the following vector: $\bar{A} \times \bar{B}$ where, $\bar{A}$ = (-2, -5, 2) $\bar{B}$ = (-5, -2, -3)
$[(-2, -5, 2) \times (-5, -2, -3)] =(19, -16, -21)$ $\mathrm{Magnitude} = 23 \sqrt(2) = 32.5269$