Cross product of vectors

The vector product or cross product of any two vectors and denoted as (read cross ) is denoted as :

, here θ is the angle between the vectors and . gives the direction of the product vector. Its direction is perpendicular to the plane containing the two vectors and follows the right-hand-screw rule or right-hand-thumb rule.

Right-Hand-Thumb Rule :

To find the direction of , drew the two vectors and with both the tails coinciding. Now place your stretched right palm perpendicular to the plane of and in such a way that the figures are along the vector and when the figures are folded they go towards , then the direction of the thumb gives the direction of .

Properties of cross product

Cross product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. perpendicular to both the vectors and , through the vectors and may or may not be orthogonal.
Vector product of two vectors is not commutative i.e.

Although, , their direction are different (opposite).

3. The vector product is distributive when the order of the vectors is strictly maintained i.e.

4. Cross product of a vector to itself is a null vector.

5. In case of unit vector ,

In case of orthogonal unit vectors , and in accordance with right-hand-thumb-rule,

, and

Similarly, if the order of the unit vectors are changed

, ,

6. In terms of components,

Example 1. is East wards and is downwards. Find the direction of ?

Solution . Applying right hand thumb rule we find that is along North.

Example 2. if , find angle between and ,

Solution. or AB cos θ = AB sin θ , tan θ = 1 ⇒ θ =45º