Jacobi Identity

Let $\mathbf{a, b, c} \in \mathbb{R}^3$. Prove the following identity:

$$\mathbf{a}\left( \mathbf{b}\times \mathbf{c} \right)+\mathbf{b} \left( \mathbf{c} \times \mathbf{a} \right) + \mathbf{c} \left( \mathbf{a} \times \mathbf{b} \right)=0$$

which is known as Jacobi's identity and give a geometrical interpretation.

Solution

The identity follows immediately using the following facts:

• $\mathbf{a} \times \left( \mathbf{b} \times \mathbf{c}\right) = (\mathbf{a}\cdot \mathbf{c}) \mathbf{b} - (\mathbf{a}\cdot \mathbf{b})\mathbf{c}$ (this vector falls in the plane spanned by $\mathbf{b, c}$)
• $\mathbf{b}\times \left( \mathbf{c} \times \mathbf{a} \right) = (\mathbf{b}\cdot \mathbf{a}) \times \mathbf{c} - (\mathbf{b}\cdot \mathbf{c})\mathbf{a}$ (this vector falls in the plane spanned by $\mathbf{c, a}$)
• $\mathbf{c}\times \left( \mathbf{a} \times \mathbf{b} \right) = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a})\mathbf{b}$ (this vector falls in the plane spanned by $\mathbf{b, a}$)

 Adding these relations by part we get the identity.

Well, here is a visualization. Imagine these in terms of three planes which intersect along the directions $\mathbf{a, b, c}$.

The idea is that the lengths of the orange, purple and cyan arrows are indicative of the dot-products which appear in the spans.