Vector or cross product: The vector product of two vectors A and B, is a vector which is defined as

A x B = AB sinθ n̂               ………..                   (2.22)

Where n̂ is a unit vector perpendicular to the plane containing A and B as shown in fig. 2.12 (a). its direction can be determined by right hand rule. For that purpose place together the tails of vectors A and B to define the plane of vectors A and B. the direction of the product vector is perpendicular to this plane. Rotate the first vector A into B through the smaller of the two possible angles and curl the fingers of the right hand in the direction of rotation, keeping the thumb erect. The direction of the product vector will be along the erect thumb, as shown in the fig 2.12 (b). because of this direction rule, B x A is a vector opposite in sign to A x B. hence,

plane_containing_A and B

Fig. 2.12 (a)

 

erect_thumb

Fig. 2.12 (b)

A x B = -B x A      ……………..             (2.23)

vector_or_cross_product

Fig. 2.12 (c)

 

Characteristics of cross product

  1. Since A x B is not the same as B x A, the cross product is non commutative.
  2. The cross product of two perpendicular vectors has maximum magnitude A x B = AB sin90⁰ n̂ = AB in case of unit vectors, since they form a right handed system and are mutually perpendicular fig. 2.5 (a)

Î x ĵ = k̂ , ĵ x k̂ = î , k̂xî = ĵ

  1. The cross product of two parallel vectors is null vector, because for such vectors θ = 0⁰ or 180⁰. Hence

A x B = AB sin0⁰n̂ = AB sin 180⁰n̂ = 0

As a consequence A x A = 0

Also       î x î = ĵ x ĵ = k x k = 0  ……………..    (2.24)

  1. Cross product of two vectors A and B in terms of their rectangular components is:

A x B = (AxÎ + Ayĵ+Azk̂)x(Bxî +By ĵ+Bzk̂)

A x B = (AyBzBy) Î + (AzBx – AxBz) Ĵ+(AxBy-AyBx) k̂………..(2.25)

The result obtained can be expressed for memory in determinant form as below:

A x B =

I            j           k

Ax       Ay         Az

Bx       By         Bz

 

 

 

vector_or_cross_product_B x A

Fig. 2.12 (d)

  1. The magnitude of A x B is equal to the area of the parallelogram formed with A and B as two adjacent sides (Fig. 2.12 d).

Examples of vector product

  1. When a force F is applied on a rigid body at a point whose position vector is r from any point of the axis about which the body rotates, then the turning effect of the force, called the torque

τ = r x F

  1. The force on a particle of charge q and velocity v in a magnetic field of strength B is given by vector product.

F = q (v x B)