Vector Product of Two Vectors

Vector Product of Two Vectors

The physical quantities may be broadly classified into the vectors and the scalars. The quantities with magnitude and direction both are known as vector quantities.

What is a vector product?

The vector product or the cross product of two vectors is a binary operation on two vectors in three dimensional spaces and the vector product is denoted by x and the product of two vectors is a vector.

The vector product or cross product of two vectors A and B is a vector C, defined as: C = A x B.

We can find the Cartesian components of C = A x B in terms of the components of A and B.

C_x = A_yB_z – A_zB_y

C_y = A_zB_x – A_xB_z

C_z = A_xB_y – A_yB_x

Again, consider two arbitrary vectors A and B and choose the orientation of your Cartesian coordinate system such that A point into the x – direction and B lies in the x – y plane. Then,

A = (A_x, 0, 0); B = (B_x, B_y, 0) and C_x = 0.

C_y = 0

C_z = A_xB_y

Then, the magnitude of C is,

C = C_z = A_xB_y

Since, A_x = A and B_y = B sinΦ we can write it as, C = AB sinΦ.

Where,

Φ = smallest angle between the directions of the vectors A and B.

C is perpendicular to both A and B.

i.e., it is perpendicular to the plane that contains both A and B. the direction of C can be found by inspecting its components or by using Right – Hand rule.

Let the fingers of your right hand point in the direction of A.  Orient the palm of your hand so that, as you curl your fingers, you can sweep them over to point in the direction of B.  Your thumb points in the direction of C = A X B.

If A and B are parallel or anti – parallel to each other, then C = A x B = 0. Since, sinΦ = 0. If A and B are perpendicular to each other, then sinΦ = 1 and C has its maximum possible magnitude.

When we form the scalar product of two vectors, we multiply the perpendicular component of the two vectors. The vector product is not commutative.

A x B = – B x A.