Vector product of two vectors

The vector (cross) product c = a × b has magnitude ab sin θ, where θ is the smaller angle between a and b, and points perpendicular to the plane of a and b.

Direction by the right-hand rule: curl the fingers of the right hand from a to b through the smaller angle; the thumb gives c. A right-handed screw turned from a to b advances along c.

The cross product is not commutative: b × a = −(a × b), same size, opposite direction. The scalar product is commutative.

Under reflection every component changes sign, so a → −a and b → −b, but a × b is unchanged.

Both products are distributive: a × (b + c) = a × b + a × c. Also a × a = 0.

Unit vectors: i × i = j × j = k × k = 0; i × j = k, j × k = i, k × i = j, and the products in the reverse order are negative.

In components, a × b = (a_y b_z − a_z b_y) i + (a_z b_x − a_x b_z) j + (a_x b_y − a_y b_x) k, which is the determinant with rows (i, j, k), (aₓ, a_y, a_z), (bₓ, b_y, b_z).

For a = 3i − 4j + 5k and b = −2i + j − 3k: a · b = −25 and a × b = 7i − j − 5k, so b × a = −7i + j + 5k.