c×a

c×a denotes the cross product of two vectors, typically written c × a. The operation is defined for two vectors in three-dimensional space and yields a new vector perpendicular to both input vectors. The magnitude of c × a equals |c||a|sinθ, where θ is the angle between c and a, and the direction is given by the right-hand rule.

If c = (c1, c2, c3) and a = (a1, a2, a3), then c × a = (c2 a3 - c3 a2, c3 a1 - c1 a3, c1 a2 - c2 a1). This coordinate form is often used for practical calculations in physics and engineering.

Key properties include bilinearity and anti-commutativity. Specifically, c × (a + b) = c × a + c × b, and c × a = - (a × c). The cross product is zero when the vectors are parallel or one is the zero vector. The resulting vector is perpendicular to the plane containing c and a.

Geometrically, the magnitude of c × a equals the area of the parallelogram spanned by c and a, and its direction is normal to that plane, oriented per the right-hand rule. This makes the cross product useful for determining rotational directions and perpendicular axes in three-dimensional space.

Applications of the cross product are widespread in physics and engineering. Examples include torque, given by τ = r × F, angular momentum L = r × p, and the magnetic force component F = q v × B. While the standard cross product is defined in three dimensions, related vector products exist in higher dimensions with restricted properties, and the operation is not generally defined beyond three dimensions in the same form.