If $\vec{A}.\vec{B}=\vec{B}.\vec{C}$, then $\vec{A}$ may not always be equal to $\vec{C}$
Reason :- the dot product of two vectors involves the cosine of the angle between the two vectors.
(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.

Hint:
In this question, we are given $\vec{A}.\vec{B}=\vec{B}.\vec{C}$. We use the dot product to find out the correct option. We know in expanding the dot product, we use the cosine of the angle between them. Hence by expanding the dot product, we get the two angles and $\vec{A}$and $\vec{C}$ will be equal only when their angles are equal. Then we are able to find out the correct option.



Complete step by step solution:
Given $\vec{A}.\vec{B}=\vec{B}.\vec{C}$
This means $AB\cos \alpha =BC\cos \beta $
That is $AB\cos \alpha =BC\cos \beta $
Where $\alpha $is the angle between vector $\vec{A}$and $\vec{B}$ and
$\beta $is the angle between vectors $\vec{B}$ and $\vec{C}$
Then A = C only when $\alpha =\beta $
Thus, it depends on the magnitude as well as the cosine of the angle between the vectors.
So when angle between $\vec{A}$ and $\vec{B}$is equal to angle between $\vec{B}$and $\vec{C}$only then
$\vec{A}$is equal to $\vec{C}$
Hence, both assertion and reason are correct and Reason is the correct explanation for assertion.
Thus, Option (A) is correct.


Therefore, the correct option is A.




Note:
 In this question, we must know about the dot vector. Dot product tells us what amount of one vector goes in the direction of another. For eg :- if you pulled a box 10 m at an inclined angle, there is a horizontal component and a vertical component to your force vector. So the dot product in this case would give you the amount of force going in the direction of displacement or in the direction that box moved.