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

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

Example

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

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 →