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Scalar Triple Product

Consider three vectors $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ . The scalar triple product is defined $ {\bf a}\cdot {\bf b}\times {\bf c}$ . Now, $ {\bf b}\times {\bf c}$ is the vector area of the parallelogram defined by $ {\bf b}$ and $ {\bf c}$ . So, $ {\bf a}\cdot {\bf b}\times {\bf c}$ is the scalar area of this parallelogram multiplied by the component of $ {\bf a}$ in the direction of its normal. It follows that $ {\bf a}\cdot {\bf b}\times {\bf c}$ is the volume of the parallelepiped defined by vectors $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ . (See Figure A.12.) This volume is independent of how the triple product is formed from $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ , except that

$\displaystyle {\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}.$ (A.53)

So, the ``volume'' is positive if $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ form a right-handed set (i.e., if $ {\bf a}$ lies above the plane of $ {\bf b}$ and $ {\bf c}$ , in the sense determined from a right-hand circulation rule by rotating $ {\bf b}$ onto $ {\bf c}$ ), and negative if they form a left-handed set. The triple product is unchanged if the dot and cross product operators are interchanged,

$\displaystyle {\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}.$ (A.54)

The triple product is also invariant under any cyclic permutation of $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ ,

$\displaystyle {\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b},$ (A.55)

but any anti-cyclic permutation causes it to change sign,

$\displaystyle {\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}.$ (A.56)

The scalar triple product is zero if any two of $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ are parallel, or if $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ are coplanar.

Figure A.12: A vector parallelepiped.
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If $ {\bf a}$ , $ {\bf b}$ , and $ {\bf c}$ are non-coplanar then any vector $ {\bf r}$ can be written in terms of them: that is,

$\displaystyle {\bf r} = \alpha \,{\bf a} + \beta\,{\bf b} + \gamma\, {\bf c}.$ (A.57)

Forming the dot product of this equation with $ {\bf b}\times {\bf c}$ , we then obtain

$\displaystyle {\bf r} \cdot {\bf b} \times{\bf c} = \alpha\, {\bf a}\cdot{\bf b} \times{\bf c},$ (A.58)

so

$\displaystyle \alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}.$ (A.59)

Analogous expressions can be written for $ \beta $ and $ \gamma$ . The parameters $ \alpha $ , $ \beta $ , and $ \gamma$ are uniquely determined provided $ {\bf a}\cdot{\bf b} \times{\bf c} \neq 0$ : that is, provided the three vectors are non-coplanar.


next up previous
Next: Vector Triple Product Up: Vectors and Vector Fields Previous: Rotation
Richard Fitzpatrick 2016-01-22