Question

Assertion :- Vector addition is commutative.
Reason :- ($\vec{A}$+ $\vec{B}$) $\ne $ ($\vec{B}$+ $\vec{A}$)
( a ) Both assertion and reason are correct and reason is the correct explanation for assertion.
( b ) Both assertion and reason are correct and reason is not the correct explanation for assertion.
( c ) Assertion is correct but the reason is incorrect.
( d ) Both assertion and reason are incorrect.

Answer 1

Hint:
In this question, we are given that the addition of a vector is commutative. And the commutative law says that in which order we add the terms doesn’t matter. That is x+y = y+x. We find the vector A + B is equal or not equal to vector B + A then we choose the correct option.


Complete step by step solution:
Consider that we have two vectors $\vec{A}$ and $\vec{B}$ and we suppose that these are in ‘n’ dimensions.
Therefore, we can write $\vec{A}$as
< ${{A}_{1}},{{A}_{2}},{{A}_{3}},.....................,{{A}_{n}}$> and
$\vec{B}$ can be written as
<${{B}_{1}},{{B}_{2}},{{B}_{3}},.....................,{{B}_{n}}$>
Now we can find out $\vec{A}$ + $\vec{B}$
That is $\vec{A}$ + $\vec{B}$ = < ${{A}_{1}}+{{B}_{1}},{{A}_{2}}+{{B}_{2}},{{A}_{3}}+{{B}_{3}},.....................,{{A}_{n}}+{{B}_{n}}$>
As all the ${{A}_{i}}'s$ and the ${{B}_{i}}'s$ are the real numbers, therefore we can write the above equation as
$\vec{A}$ + $\vec{B}$ = <${{B}_{1}}+{{A}_{1}},{{B}_{2}}+{{A}_{2}},{{B}_{3}}+{{A}_{3}},.....................,{{B}_{n}}+{{A}_{n}}$>
This can be called as $\vec{B}$+ $\vec{A}$
Since vector addition is commutative,
Therefore :- ($\vec{A}$+ $\vec{B}$) = ($\vec{B}$+ $\vec{A}$)
Hence, the assertion is correct but the reason is incorrect.
Thus, Option (C) is the correct answer.


Therefore, the correct option is C.




Note:
In this question, we have to add the two vectors. Students must keep in mind the basic properties of vectors and how these properties are implemented on vectors. Questions may be asked on other properties like additive, homogeneity etc.