### Cross product examples

#### Example 1

Calculate the cross product between $\vc{a} = (3, -3, 1)$ and $\vc{b} = (4,9,2)$.

**Solution:** The cross product is
\begin{align*}
\vc{a} \times \vc{b} &= \left|
\begin{array}{ccc}
\vc{i} & \vc{j} & \vc{k}\\
3 & -3 & 1\\
4 & 9 & 2
\end{array}
\right|\\
&= \vc{i} (-3\cdot 2 -1 \cdot 9) - \vc{j}(3\cdot 2- 1 \cdot 4)
+ \vc{k}(3 \cdot 9 + 3 \cdot 4)\\
&= -15 \vc{i} -2 \vc{j} + 39 \vc{k}
\end{align*}

#### Example 2

Calculate the area of the parallelogram spanned by the vectors $\vc{a} = (3, -3, 1)$ and $\vc{b} = (4,9,2)$.

**Solution:** The area is $\| \vc{a} \times \vc{b}\|$. Using the above
expression for the cross product, we find that the area is
$\sqrt{15^2+2^2+39^2} = 5 \sqrt{70}$.

#### Example 3

Calculate the area of the parallelogram spanned by the vectors $\vc{a} = (3,-3,1)$ and $\vc{c} = (-12, 12, -4)$.

**Solution: **
\begin{align*}
\vc{a} \times \vc{c} &= \left|
\begin{array}{ccc}
\vc{i} & \vc{j} & \vc{k}\\
3 & -3 & 1\\
-12 & 12 & -4
\end{array}
\right|\\
&= (0,0,0)
\end{align*}

The magnitude of the zero vector is zero, so the area of the parallelogram is zero. What happened?

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##### Vector algebra

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