Vector Cross Product

Formulas and examples for the cross product of two vectors

This section describes how to calculate the cross product of two vectors;

The cross product, also known as vector product, is a link in the three-dimensional Euclidean vector space that assigns a vector to two vectors. To distinguish it from other products, especially the scalar product, it is written in German and English-speaking countries with a cross \(×\) as a multiplication symbol.


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

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


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

Cross product online calculator

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