Algebraic properties and geometric interpretation

Essential properties

The vector product satisfies the following rules, for all vectors u, v, w and any real number k:

• Antisymmetry: u ^ v = -(v ^ u). Unlike the scalar product, the order matters!
• Bilinearity: (ku) ^ v = k(u ^ v) = u ^ (k*v), and u ^ (v + w) = u ^ v + u ^ w.
• Non-associativity: in general, u ^ (v ^ w) ≠ (u ^ v) ^ w.
• u ^ u = 0 (zero vector), for any vector u.
• u ^ v = 0 if and only if u and v are collinear (including when one of them is zero).

Geometric interpretation: the area

The norm ||u ^ v|| is equal to the area of the parallelogram formed by u and v. If you want the area of the triangle formed by points A, B and C, the formula is:

Area(ABC) = (1/2) * ||AB ^ AC||

Example

Let A(0,0,0), B(2,0,0) and C(0,3,0). We have AB = (2,0,0) and AC = (0,3,0).

AB ^ AC = (00 - 03, 00 - 20, 23 - 00) = (0, 0, 6)

Area(ABC) = (1/2) * ||(0,0,6)|| = (1/2) * 6 = 3, which indeed corresponds to the area of the right-angled triangle with sides 2 and 3: (1/2)23 = 3.

Common pitfall

Do not confuse antisymmetry and anticommutativity with a sign error: if you swap the two vectors in the cross product, the resulting vector changes direction (it points in the opposite direction), but retains the same magnitude. Also remember to check that the basis used is indeed direct orthonormal: the formula in coordinates does not apply as it stands in just any basis.