Dot product of cross products

Here is the formula:

$$(a \times b) \cdot (c \times d) = (a \cdot c)(b \cdot d) - (a \cdot d) (b \cdot d)$$

This can be easily derived using the laws of cross product. A interesting special case is when the two cross products are the same:

$$\begin{align*} (a \times b) \cdot (a \times b) &= (a \cdot a) (b \cdot b) - (a \cdot b)^2 \\ &= |a|^2|b|^2 - (a \cdot b)^2 \end{align*}$$

But we also have

$$\begin{align*} (a \times b) \cdot (a \times b) &= |a \times b|^2 \end{align*}$$

Hence

$$\begin{align*} |a|^2|b|^2 - (a \cdot b)^2 &= |a \times b|^2 \\ 1 - \cos^2 \theta &= \left( \frac{|a \times b|}{|a||b|} \right)^2 \\ \sin \theta &= \frac{|a \times b|}{|a||b|} \end{align*}$$

An interesting conclusion, huh?