Vector Cross Product Calculator

The calculator on this page calculates the vector cross product for vectors with 3 elements.

Enter the values of the vectors and click on the 'Calculate' button.

The value 0 is assumed for empty fields.

Formula

In a real coordinate space \(\displaystyle \mathbb {R}^{3} \) with the standard dot product and standard orientation, the cross product is:

\( \vec{a}\; \times\; \vec{b} = \left[\matrix{a_1\\a_2\\a_3}\right] \times \left[\matrix{b_1\\b_2\\b_3}\right] = \left[ \matrix{a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1 } \right]\)

Example

\( \vec{a}\; \times\; \vec{b} = \left[\matrix{1\\2\\3}\right] \times \left[\matrix{7\\8\\9}\right] = \left[ \matrix{2\cdot 9 - 3\cdot 8\\3\cdot 7 - 1\cdot 9\\1\cdot 8 - 2\cdot 7 } \right] = \left[ \matrix{-6\\12\\-6} \right] \)

Calculate the cross product magnitude

\(\displaystyle |\vec{a}×\vec{b}| =\left|\left[\matrix{1\\-4\\5}\right] ×\left[\matrix{3\\5\\2}\right] \right| = \left|\left[\matrix{-33\\13\\17}\right]\right| \)

\(\displaystyle A=\sqrt{(-33^2)+13^2+17^2}\)

\(\displaystyle \;\;\;=\sqrt{1089+169+289}\)

\(\displaystyle \;\;\;=\sqrt{1547}\)

\(\displaystyle \;\;\;=39,33\)