Question
Prove the following formula (8) for the vector triple product:$$a \times(b \times c)=(a \cdot c) b-(a \cdot b) c$$
Step 1
This means that $a \times (b \times c)$ is not the same as $(a \times b) \times c$. Show more…
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(a) Prove the two triple-vector-product identities $$ (a \times b) \times c=(a \cdot c) b-(b \cdot c) a $$ and a $\times(b \times c)=(a \cdot c) b-(a \cdot b) c$ (b) Prove $(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=\mathbf{u} \times(\mathbf{v} \times \mathbf{w})$ if and only if $(u \times w) \times v=0$ (c) Also prove that $(\mathbf{a} \times \mathbf{v}) \times \mathbf{w}+(\mathbf{v} \times \mathbf{w}) \times \mathbf{u}+(\mathbf{w} \times \mathbf{u}) \times \mathbf{v}=\mathbf{0}$ (called the Jacobi identity).
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Use a parallelepiped to show that $\vec{a} \cdot(\vec{b} \times \vec{c})=(\vec{a} \times \vec{b}) \cdot \vec{c}$ for any vectors $\vec{a}, \vec{b},$ and $\vec{c}$
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