Question

What is commutative law?

Answer 1

Hint: Commutative law refers to the operation on the numbers where the order of the numbers can be altered. There are two laws relating to commutative on the operations of addition and multiplication, that is, \[a+b=b+a\] and \[ab=ba\] respectively. And we will see examples for these commutative addition and multiplication for better understanding.

Complete step-by-step solution:
According to the question given to us, we have to write about the commutative law.
Commutative law involves operations on numbers and it follows that the order in which the numbers are taken for the specified operation can be altered and it will still give the same result. The operation within commutative law are:
1) Commutative law of addition –
The numbers when added up does not depend on the order of the numbers used in the operation. We have,
\[a+b=b+a\]
Here, the order can be altered and we will still get the same result.
For example – addition of two numbers 2 and 3
On the LHS side, we have, \[2+3=5\]
And on the RHS side, we have, \[3+2=5\]
That is, we can write, \[2+3=3+2=5\].
2) Commutative law of multiplication –
The numbers when multiplied are not dependent on the order of the numbers involved in the operation, rather the order can be altered.
We have,
\[ab=ba\]
Here, we are using the scalar multiplication or the so-called dot product.
For example – multiplication of two numbers 2 and 3
On the LHS side, we have, \[2\cdot 3=6\]
And on the RHS side, we have, \[3\cdot 2=6\]
\[LHS=RHS\], so we can write \[2\cdot 3=3\cdot 2=6\]
Therefore, the commutative laws are: \[a+b=b+a\] and \[ab=ba\].

Note: In the commutative law of multiplication, we implemented scalar multiplication or the so-called dot product. So, dot product follows commutative law. But the vector multiplication or the cross product does not adhere to the commutative law as it gives \[a\times b=-b\times a\]. Therefore, vector multiplication is not commutative in nature.