Question

Assertion
\[\begin{array}{*{20}{c}}
  {\overrightarrow A .\overrightarrow B }& = &{\overrightarrow B .\overrightarrow A }
\end{array}\]
Reason
Dot product two vectors is commutative.
A) Both assertion and reason are correct and Reason is the correct explanation for the assertion.
B) Both assertion and reason are correct and the reason is not the correct explanation for the assertion.
C) Assertion is correct but Reason is incorrect.
D) Both assertion and reason are incorrect.

Answer 1

Hint:
In this question, we have given that the dot product of two vectors is commutative. First of all, determine the dot product of the two vectors. The dot product of two vectors is the product of the magnitude of these vectors and the angle between them. And then, we will select the correct answer from the given option.


Complete step by step solution:
Let us assume that there are two vectors \[\overrightarrow A \]and \[\overrightarrow B \]. And the angle between them is \[\theta \]. Therefore, we know that the dot product of the two vectors is the product of the magnitude of both the vectors and the cosine angle between them.
Therefore, the dot product of the vector is \[\overrightarrow A \]and \[\overrightarrow B \]is,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \overrightarrow A .\overrightarrow B }& = &{AB\cos \theta }
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow \cos \theta }& = &{\frac{{\overrightarrow A .\overrightarrow B }}{{AB}}}
\end{array}\]………… (1)
And the dot product of \[\overrightarrow B \]and \[\overrightarrow A \]is,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \overrightarrow B .\overrightarrow A }& = &{BA\cos \theta }
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow \cos \theta }& = &{\frac{{\overrightarrow B .\overrightarrow A }}{{BA}}}
\end{array}\]………… (2)
Now the equation (1) and (2) will give,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \overrightarrow A .\overrightarrow B }& = &{\overrightarrow B .\overrightarrow A }
\end{array}\].
Therefore, from the result, we can conclude that the vector product is commutative. So, the assertion and reason both are true and the reason is the correct explanation for the assertion.
Therefore, the correct option is A.




Note:
In this problem, we have given that the dot product of two vectors is commutative. Commutative law says that the resultant of two vectors remains the same if the vectors are multiplied in any order.