Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be vectors in the space $\mathbb{E}^3$. Then it’s easy to check by calculation that the following equation holds: $$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\cdot \mathbf{c}) \mathbf{b}-(\mathbf{a}\cdot \mathbf{b}) \mathbf{c}.$$
However, the textbook says the equation can be explained geometrically, and I wonder how. All that has come to me so far is that the vector $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})$ lies on the plane spanned by $\mathbf{b},\mathbf{c}$ (given that they are not parallel), but what exactly can represent the double cross product geometrically? Please help.
The absolute value of the product may be written as $|a×(b×c)|=|a||b×c|\sin\phi,$ where $\phi$ is the deviation between $a$ and $b×c.$
Since the segment with length $|a|\sin\phi$ is orthogonal to the one with length $|b×c|,$ you may interpret the product as describing a rectangle with sides $a\sin\phi,\,b×c,$ and the absolute value of the product gives its area.