Equivalent Expressions for Scalar Triple Product

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Theorem

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

\(\ds \mathbf a\) \(=\) \(\ds a_i \mathbf i + a_j \mathbf j + a_k \mathbf k\)
\(\ds \mathbf b\) \(=\) \(\ds b_i \mathbf i + b_j \mathbf j + b_k \mathbf k\)
\(\ds \mathbf c\) \(=\) \(\ds c_i \mathbf i + c_j \mathbf j + c_k \mathbf k\)


Then this identity applies to the scalar triple product:

\(\ds \) \(\) \(\ds \sqbrk {\mathbf a, \mathbf b, \mathbf c} = \sqbrk {\mathbf b, \mathbf c, \mathbf a} = \sqbrk {\mathbf c, \mathbf a, \mathbf b}\)
\(\ds \) \(=\) \(\ds \mathbf a \cdot \paren {\mathbf b \times \mathbf c} = \mathbf b \cdot \paren {\mathbf c \times \mathbf a} = \mathbf c \cdot \paren {\mathbf a \times \mathbf b}\)
\(\ds \) \(=\) \(\ds \paren {\mathbf a \times \mathbf b} \cdot \mathbf c = \paren {\mathbf b \times \mathbf c} \cdot \mathbf a = \paren {\mathbf c \times \mathbf a} \cdot \mathbf b\)


while:

\(\ds \) \(\) \(\ds \sqbrk {\mathbf a, \mathbf c, \mathbf b} = \sqbrk {\mathbf b, \mathbf a, \mathbf c} = \sqbrk {\mathbf c, \mathbf b, \mathbf a}\)
\(\ds \) \(=\) \(\ds \mathbf a \cdot \paren {\mathbf c \times \mathbf b} = \mathbf b \cdot \paren {\mathbf a \times \mathbf c} = \mathbf c \cdot \paren {\mathbf b \times \mathbf a}\)
\(\ds \) \(=\) \(\ds \paren {\mathbf a \times \mathbf c} \cdot \mathbf b = \paren {\mathbf b \times \mathbf a} \cdot \mathbf c = \paren {\mathbf c \times \mathbf b} \cdot \mathbf a\)
\(\ds \) \(=\) \(\ds -\sqbrk {\mathbf a, \mathbf b, \mathbf c} = -\sqbrk {\mathbf b, \mathbf c, \mathbf a} = -\sqbrk {\mathbf c, \mathbf a, \mathbf b}\)


Proof

\(\ds \mathbf a \cdot \paren {\mathbf b \times \mathbf c}\) \(=\) \(\ds \begin {vmatrix}

a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end {vmatrix}\)

Definition of Scalar Triple Product
\(\ds \) \(=\) \(\ds -\begin {vmatrix}

b_i & b_j & b_k \\ a_i & a_j & a_k \\ c_i & c_j & c_k \\ \end {vmatrix}\)

Determinant with Rows Transposed
\(\ds \) \(=\) \(\ds \begin {vmatrix}

b_i & b_j & b_k \\ c_i & c_j & c_k \\ a_i & a_j & a_k \\ \end {vmatrix}\)

Determinant with Rows Transposed
\(\ds \) \(=\) \(\ds \mathbf b \cdot \paren {\mathbf c \times \mathbf a}\) Definition of Scalar Triple Product
\(\ds \) \(=\) \(\ds -\begin {vmatrix}

c_i & c_j & c_k \\ b_i & b_j & b_k \\ a_i & a_j & a_k \\ \end {vmatrix}\)

Determinant with Rows Transposed
\(\ds \) \(=\) \(\ds \begin {vmatrix}

c_i & c_j & c_k \\ a_i & a_j & a_k \\ b_i & b_j & b_k \\ \end {vmatrix}\)

Determinant with Rows Transposed
\(\ds \) \(=\) \(\ds \mathbf c \cdot \paren {\mathbf a \times \mathbf b}\) Definition of Scalar Triple Product


The remaining identities follow from Dot Product Operator is Commutative.

$\blacksquare$


Sources

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