Question 6: Define the cross product (vector product) of two vectors. What geometric interpretation the cross product have? Give examples.
VECTOR OR CROSS PRODUCT
A vector product of two vectors is one which yields a vector quantity. Mathematically,
Thus if A and B are two vectors to be multiplied, then there vector or cross product is a vector, say R.
The product vector is perpendicular to both the vectors that are multiplied and has a magnitude equal to the product of their magnitudes and the sine of the smaller angle between them.
Direction of the product vector is, though perpendicular to the plane of the vectors to be multiplied, however the exact direction (for example, up or down, or rightwards or leftward) is decided by the Right Hand Rule.
Right Hand Rule: Rotate the fingers of your right hand from the side of the vector comes first in the product to the one which comes second in the product. Direction of the extended thumb will be the direction of the product vector. See the following diagram.
Vector product is anti-commutative
This is important to note that unlike dot product, cross product is not commutative. The order of the vectors in the product is important.
It is clear in the above figure and the equation of cross product that the difference comes in the direction of the product vectors. For example in the above diagram, the product vector of
points outside the paper. On the other hand product vector of has direction inside the paper, i-e, in the anti-parallel direction. Therefore, we can say the vector product is anti-commutative.
Physical significance of vector product
Vector product has key importance in the study of Physics. Important physical quantities are determined with the help of vector product.
- Torque is determined with the help of vector product of r and F .
- Angular momentum of a body is determined by vector product. where p is the linear momentum of the particle.
Apart from Mechanics, vector product is also used in other branches of Physics.
Geometrical interpretation of cross product
Geometrical interpretation of the vector product is that area of a parallelogram is determined by the use of vector product. If A and B are considered to be the adjacent sides of a parallelogram and C is the area of the parallelogram, then
Here n̂ gives the direction of the area of the parallelogram.