Triple product

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where '''B''' &times; '''C''' is the [[cross product]] of two vectors (resulting into a vector) and the dot indicates the [[inner product]] between two vectors (a scalar).
where '''B''' &times; '''C''' is the [[cross product]] of two vectors (resulting into a vector) and the dot indicates the [[inner product]] between two vectors (a scalar).
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The triple product is sometimes called the ''scalar triple product'' to distinguish it from the ''vector triple product'' '''A'''&times;('''B'''&times;'''C'''). The scalar triple product is often written as ['''A''' '''B''' '''C'''].
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==Explanation==
==Explanation==
Let '''n''' be a unit normal to the parallelogram spanned by '''B''' and '''C''' (see figure).  Let ''h'' be the height of the terminal point of the vector '''A''' above the base of the parallelepiped. Recall:
Let '''n''' be a unit normal to the parallelogram spanned by '''B''' and '''C''' (see figure).  Let ''h'' be the height of the terminal point of the vector '''A''' above the base of the parallelepiped. Recall:

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Parallelepiped spanned by vectors A, B, and C (shown in red).
Parallelepiped spanned by vectors A, B, and C (shown in red).

In analytic geometry, a triple product is a common term for a product of three vectors A, B, and C leading to a scalar (a number). The absolute value of this scalar is the volume V of the parallelepiped spanned by the three vectors:

V = \big|\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\big|,

where B × C is the cross product of two vectors (resulting into a vector) and the dot indicates the inner product between two vectors (a scalar).

The triple product is sometimes called the scalar triple product to distinguish it from the vector triple product A×(B×C). The scalar triple product is often written as [A B C].

Explanation

Let n be a unit normal to the parallelogram spanned by B and C (see figure). Let h be the height of the terminal point of the vector A above the base of the parallelepiped. Recall:

Volume V of parallelepiped is height h times area S of the base.

Note that h is the projection of A on n and that the area S is the length of the cross product of the vectors spanning the base,

h = \mathbf{A}\cdot\mathbf{n}\quad\hbox{and}\quad S = | \mathbf{B}\times\mathbf{C}|.

Use

V = (\mathbf{A}\cdot \mathbf{n})\;(| \mathbf{B}\times\mathbf{C}|) = \mathbf{A}\cdot (\mathbf{n}\,| \mathbf{B}\times\mathbf{C}|) = \mathbf{A}\cdot (\mathbf{B}\times\mathbf{C}),

where it is used that

\mathbf{n}\; |\mathbf{B}\times\mathbf{C}| = \mathbf{B}\times\mathbf{C}.

(The unit normal n has the direction of the cross product B × C).

If A, B, and C do not form a right-handed system, A•n < 0 and we must take the absolute value: | A• (B×C)|.

Triple product as determinant

Take three orthogonal unit vectors i , j, and k and write

\mathbf{A} = A_1\mathbf{i}+ A_2\mathbf{j}+A_3\mathbf{k}, \quad \mathbf{B} = B_1\mathbf{i}+ B_2\mathbf{j}+B_3\mathbf{k},\quad \mathbf{C} = C_1\mathbf{i}+ C_2\mathbf{j}+C_3\mathbf{k}.

The triple product is equal to a 3 × 3 determinant

\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = \begin{vmatrix} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \\ \end{vmatrix}.

Indeed, writing the cross product as a determinant we find

\begin{align} \mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) &= \mathbf{A}\cdot \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \\ \end{vmatrix} \\ &= \big(A_1\mathbf{i}+ A_2\mathbf{j}+A_3\mathbf{k}\big)\cdot \big[ (B_2\,C_3 - B_3\,C_2)\;\mathbf{i}+ (B_3\,C_1 - B_1\,C_3)\;\mathbf{j} + (B_1\,C_2 - B_2\,C_1)\;\mathbf{k} \big] \\ &= A_1\;(B_2\,C_3 - B_3\,C_2)+ A_2\;(B_3\,C_1 - B_1\,C_3) + A_3\;(B_1\,C_2 - B_2\,C_1)\\ &=\begin{vmatrix} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \\ \end{vmatrix} . \end{align}

Since a determinant is invariant under cyclic permutation of its rows, it follows

\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = \mathbf{B}\cdot(\mathbf{C}\times\mathbf{A}) = \mathbf{C}\cdot(\mathbf{A}\times\mathbf{B}).

Reference

M. R. Spiegel, Theory and Problems of Vector Analysis, Schaum Publishing, New York (1959) p. 26

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