The Commutative Law and the Cross Product of Vectors

The commutative law states that for any two elements, the order of the operation does not affect the result. In the case of vectors, this means that the order in which we perform the cross product should not affect the outcome. However, the correct answer to the given question is option 'B' - False. Let's understand why.

Explanation:

1. The Cross Product:
The cross product is an operation that combines two vectors to produce a third vector that is perpendicular to both of the original vectors. It is denoted by the symbol '×' and is defined as:

P × Q = ||P|| ||Q|| sin(θ) n

where P and Q are the vectors being crossed, ||P|| and ||Q|| are their magnitudes, θ is the angle between them, and n is a unit vector perpendicular to the plane formed by P and Q.

2. Commutative Law:
The commutative law states that for any two elements, the order of the operation does not affect the result. In mathematical terms, it can be written as:

a × b = b × a

This law holds true for many mathematical operations, such as addition and multiplication. However, it does not hold true for the cross product of vectors.

3. Non-Commutativity of Cross Product:
The cross product of vectors is not commutative, which means that changing the order of the vectors being crossed will result in a different outcome.

To see this, let's consider two vectors P and Q. The cross product of P and Q is given by:

P × Q = ||P|| ||Q|| sin(θ) n

Now, if we switch the order and calculate the cross product of Q and P, we get:

Q × P = ||Q|| ||P|| sin(θ') n'

Here, θ' is the angle between Q and P, and n' is a unit vector perpendicular to the plane formed by Q and P.

Since the angles and magnitudes may be different for P and Q, the resulting cross products will also be different. Thus, the commutative law does not hold for the cross product of vectors.

Conclusion:

In conclusion, the commutative law is not valid for the cross product of two vectors. The order of the vectors being crossed affects the outcome, and switching the order will result in a different cross product.