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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.

Juggler
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  • Have a look at the illustration on https://en.wikipedia.org/wiki/Cross_product#Properties that compares $a\times(b\times c)$ to $(a\times b)\times c$, which decomposes the product into projections onto a certain plane. – amd Sep 27 '19 at 00:10
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    Maybe this answer will help. – Ted Shifrin Sep 27 '19 at 00:15

1 Answers1

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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.

Allawonder
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