Calculator and formula for calculating the triple product of three vectors
This function calculates the triple product of three vectors. The triple product is the scalar product of the cross product of two vectors and a third vector. It results in the oriented volume of the space spanned by the three vectors (parallelepipeds)
To calculate, enter the values of the three vectors, then click on the 'Calculate' button
Empty fields are evaluated as 0.
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The scalar triple product is defined as the dot product of one of the vectors with the cross product of the other two.
1. Calculate triple product via vector cross product and scalar 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
\(\displaystyle \left[\matrix{2\\-1\\-1}\right] \cdot \left[\matrix{6\\0\\-2}\right] = 2\cdot 6 + (-1)\cdot 0 +(-1)\cdot(-2)\) \(\displaystyle = 12 +0+2=14\)
2. The triple product can also be calculated using the determinant of a matrix
\(\displaystyle D=\left[\matrix{a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3} \right]\)
\(\displaystyle D=\left|\matrix{1&2&6\\1&1&0\\1&3&-2}\right|\)
\(\displaystyle V= 1\cdot1\cdot(-2)+2\cdot0\cdot1 +6\cdot1\cdot3\) \(\displaystyle + 6\cdot1\cdot1 -1\cdot0\cdot3 -2\cdot1\cdot(-2)=14\)
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