Formulas and examples for calculating the vector triple product

The triple product is used to calculate the volume that is spanned by three vectors.

1. Calculate triple product via cross product and dot product

\(\displaystyle triple product = (\vec{a} \times \vec{b})·\vec{c} \) \(\displaystyle = \left( \left[\matrix{a_1\\a_2\\a_3}\right] \times \left[\matrix{b_1\\b_2\\b_3}\right]\right) ·\left[\matrix{c_1\\c_2\\c_3}\right] \)


\(\displaystyle \vec{a}=\left[\matrix{1\\1\\1}\right] \; \vec{b}=\left[\matrix{2\\1\\3}\right] \;\vec{c}=\left[\matrix{6\\0\\-2}\right] \)

Calculate cross product

\(\displaystyle \;\;\; \left[\matrix{a_1\\a_2\\a_3}\right] \times \left[\matrix{b_1\\b_2\\b_3}\right] =\left[\matrix{a_2·b_3-a_3·b_2\\a_3·b_1-a_1·b_3\\a_1·b_2-a_2·b_1}\right] \)

\(\displaystyle = \left[\matrix{1\\1\\1}\right] \times \left[\matrix{2\\1\\3}\right] =\left[\matrix{1·3-1·1\\1·2-1·3\\1·1-1·2}\right] =\left[\matrix{2\\-1\\-1}\right]\)

Calculate dot product

\(\displaystyle \left[\matrix{x_1\\x_2\\x_3}\right] \cdot \left[\matrix{y_1\\y_2\\y_3}\right] \) \( = x_1\cdot y_1 + x_2\cdot y_2 +x_3\cdot y_3\)

\(\displaystyle \left[\matrix{2\\-1\\-1}\right] \cdot \left[\matrix{6\\0\\-2}\right] \) \( = 2\cdot 6 + (-1)\cdot 0 +(-1)\cdot(-2)\) \(\displaystyle = 12 +0+2=14\)

2. Calculate the triple product using a matrix

The triple product can also be calculated using the determinant of a matrix.

\(\displaystyle D=\left[\matrix{a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3} \right]\)
\(\displaystyle D=\left|\matrix{1&2&6\\1&1&0\\1&3&-2}\right|\)

\(\displaystyle V= 1\cdot1\cdot(-2)+2\cdot0\cdot1 +6\cdot1\cdot3\) \(\displaystyle + 6\cdot1\cdot1 -1\cdot0\cdot3 -2\cdot1\cdot(-2)=14\)

Calculate vector triple product online →

Vector Definition
Vector Calculation
Vector Addition
Vector Subtraction
Vector Magnitude
Vector Dot Product
Vector Cross Produkt
Vector Angle
Triple Product

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