# 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.

$\blacksquare$