Define the cross-product in space

A new product between vectors

Unlike the scalar product (u . v), which yields a number, the vector product of two vectors u and v in a vector space, denoted u ^ v (or u × v), yields a vector. This vector w = u ^ v has three characteristics:

• its direction is perpendicular to both u and v (and therefore to the plane they span);
• its direction is given by the right-hand rule (or ‘corkscrew’ rule): one turns from u towards v along the path of the smallest angle; the thumb points in the direction of w;
• its magnitude is ||u|| * ||v|| * sin(theta), where theta is the non-oriented angle between u and v (0 <= theta <= pi).

Formula using coordinates

If u = (x₁, y₁, z₁) and v = (x₂, y₂, z₂) in a direct orthonormal basis, then:

u ^ v = (y1z2 - z1y2, z1x2 - x1z2, x1y2 - y1x2)

This order is remembered using the ‘cross’ method: we write the coordinates of u followed by those of v twice in succession, and cross them by shifting them one position to the right.

Example

With u = (1, 0, 0) and v = (0, 1, 0):

u ^ v = (00 - 01, 00 - 10, 11 - 00) = (0, 0, 1)

This indeed gives a unit vector perpendicular to the (u, v) plane, in accordance with the right-hand rule.

Common pitfall

The cross product is only defined in three-dimensional space (or in seven dimensions, a very special case not covered in the syllabus). Never confuse u ^ v (a vector) with u . v (a scalar): always check what type of result the question requires.

Product	Notation	Result	Formula (norm)
Scalar	u . v	number
Vector	u ^ v	vector