Vector cross product calculator

Calculator for calculate the cross product of 3-dimensional vectors

Calculate vector cross product

This function calculates the cross product of two 3-dimensional vectors. To perform the calculation, enter the vectors and click the Calculate button. Empty fields are counted as 0.


Cross product calculator

 Input
Vector 1Vector 2Result
× =
Decimal places
 Magnitude of the cross product
Magnitude

Description and formula for the Vector cross product

In one in the real coordinate space \(\displaystyle \mathbb {R}^{3} \) with the standard scalar product and the standard orientation applies to the cross product:

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

Example

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

Magnitude of the cross product example

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

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

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

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

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