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 threedimensional 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 Englishspeaking countries with a cross \(×\) as a multiplication symbol.
Formula
In a real coordinate space \(\displaystyle \mathbb {R}^{3} \) with the standard dot product and standard orientation, the cross product applies
\( \vec{a}\; \times\; \vec{b} = \left[\matrix{a_1\\a_2\\a_3}\right] \times \left[\matrix{b_1\\b_2\\b_3}\right] = \left[ \matrix{a_2b_3a_3b_2\\a_3b_1a_1b_3\\a_1b_2a_2b_1 } \right]\)
Example
\( \vec{a}\; \times\; \vec{b} = \left[\matrix{1\\2\\3}\right] \times \left[\matrix{7\\8\\9}\right] = \left[ \matrix{2\cdot 9  3\cdot 8\\3\cdot 7  1\cdot 9\\1\cdot 8  2\cdot 7 } \right] = \left[ \matrix{6\\12\\6} \right] \)
Cross product online calculator
