Triple Products

Since the cross product of two vectors produces another vector, it is possible to string cross products together. Use the same vectors a, b, c, and d as before for these triple vector products. Note that different choices of parentheses generally produce different results. The default order of operations for products is from left to right.


$\blacktriangleright$ Evaluate

a×(b×c) = $\left(\vphantom{ 16,4,-8}\right.$16, 4, - 8$\left.\vphantom{ 16,4,-8}\right)$


$\left(\vphantom{ a\times b}\right.$a×b$\left.\vphantom{ a\times b}\right)$×c = $\left(\vphantom{ 8,-4,-16}\right.$8, - 4, - 16$\left.\vphantom{ 8,-4,-16}\right)$


a×$\left(\vphantom{ \left( b\times c\right) \times d}\right.$$\left(\vphantom{ b\times c}\right.$b×c$\left.\vphantom{ b\times c}\right)$×d$\left.\vphantom{ \left( b\times c\right) \times d}\right)$ = $\left(\vphantom{ 0,0,0}\right.$0, 0, 0$\left.\vphantom{ 0,0,0}\right)$


$\left(\vphantom{ a\times \left( b\times c\right) }\right.$a×$\left(\vphantom{ b\times c}\right.$b×c$\left.\vphantom{ b\times c}\right)$$\left.\vphantom{ a\times \left( b\times c\right) }\right)$×d = $\left(\vphantom{ -8,-16,-24}\right.$ -8, - 16, - 24$\left.\vphantom{ -8,-16,-24}\right)$


$\left(\vphantom{ \left[ \begin{array}{ccc} 1 & -2 & 5 \end{array} \right] \times \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] }\right.$$\left[\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right.$$\begin{array}{ccc} 1 & -2 & 5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right.$$\begin{array}{ccc} 5 & 3 & -5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right]$$\left.\vphantom{ \left[ \begin{array}{ccc} 1 & -2 & 5 \end{array} \right] \times \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] }\right)$×$\left[\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right.$$\begin{array}{ccc} -7 & 2 & 8 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right]$ = $\left[\vphantom{ \begin{array}{ccc} 214 & -51 & 200 \end{array} }\right.$$\begin{array}{ccc} 214 & -51 & 200 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 214 & -51 & 200 \end{array} }\right]$


$\left[\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right.$$\begin{array}{ccc} 1 & -2 & 5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right]$×$\left[\vphantom{ \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] \times \left[ \begin{array}{ccc} -7 & 2 & 8 \end{array} \right] }\right.$$\left[\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right.$$\begin{array}{ccc} 5 & 3 & -5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right.$$\begin{array}{ccc} -7 & 2 & 8 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right]$$\left.\vphantom{ \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] \times \left[ \begin{array}{ccc} -7 & 2 & 8 \end{array} \right] }\right]$ = $\left[\vphantom{ \begin{array}{ccc} -37 & 139 & 63 \end{array} }\right.$$\begin{array}{ccc} -37 & 139 & 63 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -37 & 139 & 63 \end{array} }\right]$

To obtain intermediate results, select a portion of the expression and press the CTRL key down while evaluating. This technique does an in-place computation.

$\blacktriangleright$ CTRL + Evaluate, Evaluate

a×(b×c) = a×$\left(\vphantom{ 2,-4,2}\right.$2, - 4, 2$\left.\vphantom{ 2,-4,2}\right)$ = $\left(\vphantom{ 16,4,-8}\right.$16, 4, - 8$\left.\vphantom{ 16,4,-8}\right)$


$\left(\vphantom{ a\times b}\right.$a×b$\left.\vphantom{ a\times b}\right)$×c = $\left(\vphantom{ -2,4,-2}\right.$ -2, 4, - 2$\left.\vphantom{ -2,4,-2}\right)$×c = $\left(\vphantom{ 8,-4,-16}\right.$8, - 4, - 16$\left.\vphantom{ 8,-4,-16}\right)$


$\left(\vphantom{ \left[ \begin{array}{ccc} 1 & -2 & 5 \end{array} \right] \times \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] }\right.$$\left[\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right.$$\begin{array}{ccc} 1 & -2 & 5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right.$$\begin{array}{ccc} 5 & 3 & -5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right]$$\left.\vphantom{ \left[ \begin{array}{ccc} 1 & -2 & 5 \end{array} \right] \times \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] }\right)$×$\left[\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right.$$\begin{array}{ccc} -7 & 2 & 8 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right]$ = $\left[\vphantom{ \begin{array}{ccc} -5 & 30 & 13 \end{array} }\right.$$\begin{array}{ccc} -5 & 30 & 13 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -5 & 30 & 13 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right.$$\begin{array}{ccc} -7 & 2 & 8 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right]$ = $\left[\vphantom{ \begin{array}{ccc} 214 & -51 & 200 \end{array} }\right.$$\begin{array}{ccc} 214 & -51 & 200 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 214 & -51 & 200 \end{array} }\right]$


$\left[\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right.$$\begin{array}{ccc} 1 & -2 & 5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right]$×$\left[\vphantom{ \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] \times \left[ \begin{array}{ccc} -7 & 2 & 8 \end{array} \right] }\right.$$\left[\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right.$$\begin{array}{ccc} 5 & 3 & -5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 5 & 3 & -5 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right.$$\begin{array}{ccc} -7 & 2 & 8 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -7 & 2 & 8 \end{array} }\right]$$\left.\vphantom{ \left[ \begin{array}{ccc} 5 & 3 & -5 \end{array} \right] \times \left[ \begin{array}{ccc} -7 & 2 & 8 \end{array} \right] }\right]$ = $\left[\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right.$$\begin{array}{ccc} 1 & -2 & 5 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 1 & -2 & 5 \end{array} }\right]$×$\left[\vphantom{ \begin{array}{ccc} 34 & -5 & 31 \end{array} }\right.$$\begin{array}{ccc} 34 & -5 & 31 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 34 & -5 & 31 \end{array} }\right]$ = $\left[\vphantom{ \begin{array}{ccc} -37 & 139 & 63 \end{array} }\right.$$\begin{array}{ccc} -37 & 139 & 63 \end{array}$$\left.\vphantom{ \begin{array}{ccc} -37 & 139 & 63 \end{array} }\right]$

$\blacktriangleright$ CTRL + Evaluate, CTRL + Evaluate, Evaluate

a×$\left(\vphantom{ \left( b\times c\right) \times d}\right.$$\left(\vphantom{ b\times c}\right.$b×c$\left.\vphantom{ b\times c}\right)$×d$\left.\vphantom{ \left( b\times c\right) \times d}\right)$ = a×$\left(\vphantom{ \left( 2,-4,2\right) \times d}\right.$$\left(\vphantom{ 2,-4,2}\right.$2, - 4, 2$\left.\vphantom{ 2,-4,2}\right)$×d$\left.\vphantom{ \left( 2,-4,2\right) \times d}\right)$ = a×$\left(\vphantom{ 2,4,6}\right.$2, 4, 6$\left.\vphantom{ 2,4,6}\right)$ = $\left(\vphantom{ 0,0,0}\right.$0, 0, 0$\left.\vphantom{ 0,0,0}\right)$


$\left(\vphantom{ a\times \left( b\times c\right) }\right.$a×$\left(\vphantom{ b\times c}\right.$b×c$\left.\vphantom{ b\times c}\right)$$\left.\vphantom{ a\times \left( b\times c\right) }\right)$×d = $\left(\vphantom{ a\times \left( 2,-4,2\right) }\right.$a×$\left(\vphantom{ 2,-4,2}\right.$2, - 4, 2$\left.\vphantom{ 2,-4,2}\right)$$\left.\vphantom{ a\times \left( 2,-4,2\right) }\right)$×d = $\left(\vphantom{ 16,4,-8}\right.$16, 4, - 8$\left.\vphantom{ 16,4,-8}\right)$×d = $\left(\vphantom{ -8,-16,-24}\right.$ -8, - 16, - 24$\left.\vphantom{ -8,-16,-24}\right)$


$\left(\vphantom{ a\times b}\right.$a×b$\left.\vphantom{ a\times b}\right)$×$\left(\vphantom{ c\times d}\right.$c×d$\left.\vphantom{ c\times d}\right)$ = $\left(\vphantom{ -2,4,-2}\right.$ -2, 4, - 2$\left.\vphantom{ -2,4,-2}\right)$×$\left(\vphantom{ 1,2,-7}\right.$1, 2, - 7$\left.\vphantom{ 1,2,-7}\right)$ = $\left(\vphantom{ -24,-16,-8}\right.$ -24, - 16, - 8$\left.\vphantom{ -24,-16,-8}\right)$


$\left(\vphantom{ \left( a\times b\right) \times c}\right.$$\left(\vphantom{ a\times b}\right.$a×b$\left.\vphantom{ a\times b}\right)$×c$\left.\vphantom{ \left( a\times b\right) \times c}\right)$×d = $\left(\vphantom{ \left( -2,4,-2\right) \times c}\right.$$\left(\vphantom{ -2,4,-2}\right.$ -2, 4, - 2$\left.\vphantom{ -2,4,-2}\right)$×c$\left.\vphantom{ \left( -2,4,-2\right) \times c}\right)$×d = $\left(\vphantom{ 8,-4,-16}\right.$8, - 4, - 16$\left.\vphantom{ 8,-4,-16}\right)$×d = $\left(\vphantom{ -16,-32,0}\right.$ -16, - 32, 0$\left.\vphantom{ -16,-32,0}\right)$


$\left(\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right.$$\begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array}$$\left.\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right)$×$\left(\vphantom{ \left( \left( \begin{array}{r} .85 \\ .32 \\ -.77 \e... ...( \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} \right) }\right.$$\left(\vphantom{ \left( \begin{array}{r} .85 \\ .32 \\ -.77 \end{arra... ...( \begin{array}{r} 1.35 \\ -.23 \\ 1.26 \end{array} \right) }\right.$$\left(\vphantom{ \begin{array}{r} .85 \\ .32 \\ -.77 \end{array} }\right.$$\begin{array}{r} .85 \\ .32 \\ -.77 \end{array}$$\left.\vphantom{ \begin{array}{r} .85 \\ .32 \\ -.77 \end{array} }\right)$×$\left(\vphantom{ \begin{array}{r} 1.35 \\ -.23 \\ 1.26 \end{array} }\right.$$\begin{array}{r} 1.35 \\ -.23 \\ 1.26 \end{array}$$\left.\vphantom{ \begin{array}{r} 1.35 \\ -.23 \\ 1.26 \end{array} }\right)$$\left.\vphantom{ \left( \begin{array}{r} .85 \\ .32 \\ -.77 \end{arra... ...( \begin{array}{r} 1.35 \\ -.23 \\ 1.26 \end{array} \right) }\right)$×$\left(\vphantom{ \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} }\right.$$\begin{array}{r} -1.85 \\ .57 \\ .375 \end{array}$$\left.\vphantom{ \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} }\right)$$\left.\vphantom{ \left( \left( \begin{array}{r} .85 \\ .32 \\ -.77 \e... ...( \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} \right) }\right)$


                         = $\left(\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right.$$\begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array}$$\left.\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right)$×$\left(\vphantom{ \left( \begin{array}{c} .\,\allowbreak 226\,1 \\ -2.\,\a... ...( \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} \right) }\right.$$\left(\vphantom{ \begin{array}{c} .\,\allowbreak 226\,1 \\ -2.\,\allowbreak 110\,5 \\ -.\,\allowbreak 627\,5 \end{array} }\right.$$\begin{array}{c} .\,\allowbreak 226\,1 \\ -2.\,\allowbreak 110\,5 \\ -.\,\allowbreak 627\,5 \end{array}$$\left.\vphantom{ \begin{array}{c} .\,\allowbreak 226\,1 \\ -2.\,\allowbreak 110\,5 \\ -.\,\allowbreak 627\,5 \end{array} }\right)$×$\left(\vphantom{ \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} }\right.$$\begin{array}{r} -1.85 \\ .57 \\ .375 \end{array}$$\left.\vphantom{ \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} }\right)$$\left.\vphantom{ \left( \begin{array}{c} .\,\allowbreak 226\,1 \\ -2.\,\a... ...( \begin{array}{r} -1.85 \\ .57 \\ .375 \end{array} \right) }\right)$


                         = $\left(\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right.$$\begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array}$$\left.\vphantom{ \begin{array}{r} .35 \\ -.73 \\ 1.2 \end{array} }\right)$×$\left(\vphantom{ \begin{array}{c} -.\,\allowbreak 433\,76 \\ 1.\,\allowbreak 076\,1 \\ -3.\,\allowbreak 775\,5 \end{array} }\right.$$\begin{array}{c} -.\,\allowbreak 433\,76 \\ 1.\,\allowbreak 076\,1 \\ -3.\,\allowbreak 775\,5 \end{array}$$\left.\vphantom{ \begin{array}{c} -.\,\allowbreak 433\,76 \\ 1.\,\allowbreak 076\,1 \\ -3.\,\allowbreak 775\,5 \end{array} }\right)$ = $\left(\vphantom{ \begin{array}{c} 1.\,\allowbreak 464\,8 \\ .\,\allowbreak 800\,91 \\ .0\,\allowbreak 599\,9 \end{array} }\right.$$\begin{array}{c} 1.\,\allowbreak 464\,8 \\ .\,\allowbreak 800\,91 \\ .0\,\allowbreak 599\,9 \end{array}$$\left.\vphantom{ \begin{array}{c} 1.\,\allowbreak 464\,8 \\ .\,\allowbreak 800\,91 \\ .0\,\allowbreak 599\,9 \end{array} }\right)$

Tip    Parentheses are important. As always, careful and consistent use of mathematical notation is in order. When in doubt, add an extra set of parentheses to clarify an expression.


When mixing cross products with scalar products, use parentheses for clarity.

$\blacktriangleright$ Evaluate

(1, 0, 1)⋅$\left(\vphantom{ (1,2,3)\times (3,2,1)}\right.$(1, 2, 3)×(3, 2, 1)$\left.\vphantom{ (1,2,3)\times (3,2,1)}\right)$ =   - 8

$\left(\vphantom{ (1,0,1)\times (1,2,3)}\right.$(1, 0, 1)×(1, 2, 3)$\left.\vphantom{ (1,0,1)\times (1,2,3)}\right)$⋅(3, 2, 1) =   -8


Without these parentheses, you obtain different results for the first product.

$\blacktriangleright$ Evaluate

(1, 0, 1)⋅(1, 2, 3)×(3, 2, 1) =  4$\left[\vphantom{ \begin{array}{c} 3 \\ 2 \\ 1 \end{array} }\right.$$\begin{array}{c} 3 \\ 2 \\ 1 \end{array}$$\left.\vphantom{ \begin{array}{c} 3 \\ 2 \\ 1 \end{array} }\right]$

(1, 0, 1)×(1, 2, 3)⋅(3, 2, 1) =   - 8

(1, 2, 3)×(3, 2, 1)⋅(1, 0, 1) =   -8


Note     The triple scalar product has an interesting geometric interpretation. The volume of the parallelepiped spanned by three vectors A, B, and C is equal to $\left\vert\vphantom{ A\cdot \left( B\times C\right) }\right.$A⋅$\left(\vphantom{ B\times C}\right.$B×C$\left.\vphantom{ B\times C}\right)$$\left.\vphantom{ A\cdot \left( B\times C\right) }\right\vert$. Thus, the volume of the parallelepiped spanned by $\left(\vphantom{ 1,1,0}\right.$1, 1, 0$\left.\vphantom{ 1,1,0}\right)$, $\left(\vphantom{ 1,0,1}\right.$1, 0, 1$\left.\vphantom{ 1,0,1}\right)$, and $\left(\vphantom{ 0,1,1}\right.$0, 1, 1$\left.\vphantom{ 0,1,1}\right)$ is given by

$\displaystyle \left\vert\vphantom{ \left( 1,1,0\right) \cdot \left[ \left( 1,0,1\right) \times \left( 0,1,1\right) \right] }\right.$$\displaystyle \left(\vphantom{ 1,1,0}\right.$1, 1, 0$\displaystyle \left.\vphantom{ 1,1,0}\right)$⋅$\displaystyle \left[\vphantom{ \left( 1,0,1\right) \times \left( 0,1,1\right) }\right.$$\displaystyle \left(\vphantom{ 1,0,1}\right.$1, 0, 1$\displaystyle \left.\vphantom{ 1,0,1}\right)$×$\displaystyle \left(\vphantom{ 0,1,1}\right.$0, 1, 1$\displaystyle \left.\vphantom{ 0,1,1}\right)$$\displaystyle \left.\vphantom{ \left( 1,0,1\right) \times \left( 0,1,1\right) }\right]$$\displaystyle \left.\vphantom{ \left( 1,1,0\right) \cdot \left[ \left( 1,0,1\right) \times \left( 0,1,1\right) \right] }\right\vert$ =  2

In particular, this value does not depend on the order of the vectors in the triple scalar product.

$\displaystyle \left\vert\vphantom{ \left( 1,0,1\right) \cdot \left[ \left( 1,1,0\right) \times \left( 0,1,1\right) \right] }\right.$$\displaystyle \left(\vphantom{ 1,0,1}\right.$1, 0, 1$\displaystyle \left.\vphantom{ 1,0,1}\right)$⋅$\displaystyle \left[\vphantom{ \left( 1,1,0\right) \times \left( 0,1,1\right) }\right.$$\displaystyle \left(\vphantom{ 1,1,0}\right.$1, 1, 0$\displaystyle \left.\vphantom{ 1,1,0}\right)$×$\displaystyle \left(\vphantom{ 0,1,1}\right.$0, 1, 1$\displaystyle \left.\vphantom{ 0,1,1}\right)$$\displaystyle \left.\vphantom{ \left( 1,1,0\right) \times \left( 0,1,1\right) }\right]$$\displaystyle \left.\vphantom{ \left( 1,0,1\right) \cdot \left[ \left( 1,1,0\right) \times \left( 0,1,1\right) \right] }\right\vert$ = 2  
$\displaystyle \left\vert\vphantom{ \left( 0,1,1\right) \cdot \left[ \left( 1,1,0\right) \times \left( 1,0,1\right) \right] }\right.$$\displaystyle \left(\vphantom{ 0,1,1}\right.$0, 1, 1$\displaystyle \left.\vphantom{ 0,1,1}\right)$⋅$\displaystyle \left[\vphantom{ \left( 1,1,0\right) \times \left( 1,0,1\right) }\right.$$\displaystyle \left(\vphantom{ 1,1,0}\right.$1, 1, 0$\displaystyle \left.\vphantom{ 1,1,0}\right)$×$\displaystyle \left(\vphantom{ 1,0,1}\right.$1, 0, 1$\displaystyle \left.\vphantom{ 1,0,1}\right)$$\displaystyle \left.\vphantom{ \left( 1,1,0\right) \times \left( 1,0,1\right) }\right]$$\displaystyle \left.\vphantom{ \left( 0,1,1\right) \cdot \left[ \left( 1,1,0\right) \times \left( 1,0,1\right) \right] }\right\vert$ = 2