Cross Product

The cross product (or vector product) is an operation on 2 vectors $ \vec{u} $ and $ \vec{v} $ of 3D space (not collinear) whose result noted $ \vec{u} \times \vec{v} = \vec{w} $ (or sometimes $ \vec{u} \wedge \vec{v} $) is an orthogonal vector to the first 2 vectors.

For any pair of vectors $ \vec{u} = (u_1, u_2, u_3) $ and $ \vec{v} = (v_1, v_2, v_3) $, the calculation of the cross product is given by: $$ \vec{u} \times \vec{v} = \begin{pmatrix} u_2v_3-u_3v_2 \\ u_3v_1-u_1v_3 \\ u_1v_2-u_2v_1 \end{pmatrix} $$

The calculation of the vector product makes it possible to:

— check if 2 vectors are collinear (then their vector product is the zero vector)

— calculate a vector orthonogal to the 2 others and thus create an orthonogal basis with the 3 vectors

— check that 2 vectors are orthogonal

— calculate the area of a parallelogram with sides $ \vec{u} $ and $ \vec{v} $ (the modulus of the vector product is equal to the area of the parallelogram)

The cross product is distributive with the addition:

$$ \vec{a} \times ( \vec{b} + \vec{c} ) = \vec{a} \times \vec{b} + \vec{a} \times \

ec{c} $$

The cross product is distributive with a scalar multiplication:

$$ \lambda (\vec{a} \times \vec{b}) = \lambda \vec{a} \times \vec{b} = \vec{a} \times \lambda \ vec{b} $$

The cross product is antisymmetric:

$$ \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} $$

The norm (the modulus) of the vector product is defined by the formula:

$$ \| \vec{u} \times \vec{v} \| = \| \vec{u} \| \| \vec{v} \| \left| \sin ( \widehat{ \vec{u}, \vec{v} } ) \right| $$