Scalar Triple Product as Product of Magnitudes

Theorem

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:

\(\ds \mathbf a\)

\(=\)

\(\ds a_i \mathbf i + a_j \mathbf j + a_k \mathbf k\)

\(\ds \mathbf b\)

\(=\)

\(\ds b_i \mathbf i + b_j \mathbf j + b_k \mathbf k\)

\(\ds \mathbf c\)

\(=\)

\(\ds c_i \mathbf i + c_j \mathbf j + c_k \mathbf k\)

where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis of $\mathbf V$.

Let $\sqbrk {\mathbf a, \mathbf b, \mathbf c}$ be the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.

Then:

$\sqbrk {\mathbf a, \mathbf b, \mathbf c} = \size {\mathbf a} \size {\mathbf b} \size {\mathbf c} \sin \theta \cos \alpha$

where:

$\theta$ is the angle between $\mathbf b$ and $\mathbf c$

$\alpha$ is the angle between $\mathbf b \times \mathbf c$ and $\mathbf a$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): triple product
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): triple product