Question

Assertion
The sum of two vectors may be zero.
Reason
The vector cancels each other, when they are equal and opposite.
A) Both assertion and reason are correct and the reason is the correct clarification for the assertion.
B) Both assertion and reason are correct and the reason isn't the right clarification for the assertion.
C) Assertion is correct however the reason is wrong.
D) Both assertion and reason are wrong.

Answer 1

Hint:
In this question, we have given that if there are two vectors, then their sum may be zero. Therefore, the Sum of the two vectors may be zero only when both the vectors are opposite in direction to each other but should have the same magnitude. And then choose the correct option.


Formula used:
1) \[\begin{array}{*{20}{c}}
  R& = &{\sqrt {{{\left( A \right)}^2} + {{\left( B \right)}^2} + 2AB\cos \theta } }
\end{array}\]


Complete step by step solution:
Let us assume that there are two vectors \[\overrightarrow A \]and \[\overrightarrow B \]respectively. Vectors have an equivalent magnitude but are in the opposite direction. These vectors are on the same axis but are in opposite directions as shown in the figure. The resultant or sum of the vectors is \[R\].
Now suppose that the magnitude of both the vectors are unit.

Figure 1
Now according to the given figure 1, the angle between the vectors is \[{180^ \circ }\].
Now, we know that
\[ \Rightarrow \begin{array}{*{20}{c}}
  R& = &{\sqrt {{{\left( A \right)}^2} + {{\left( B \right)}^2} + 2AB\cos \theta } }
\end{array}\]
Therefore,
\[ \Rightarrow \begin{array}{*{20}{c}}
  R& = &{\sqrt {{{\left( 1 \right)}^2} + {{\left( 1 \right)}^2} + 2 \times 1 \times 1 \times \cos {{180}^ \circ }} }
\end{array}\]
From the trigonometric table, we know that the value of \[\cos {180^ \circ }\]is \[ - 1\].
Therefore,
\[ \Rightarrow \begin{array}{*{20}{c}}
  R& = &{\sqrt {{{\left( 1 \right)}^2} + {{\left( 1 \right)}^2} + 2 \times 1 \times 1 \times \left( { - 1} \right)} }
\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}
  R& = &{\sqrt {2 - 2} }
\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}
  R& = &0
\end{array}\].
From the above result, we can conclude that the sum of the two vectors may be zero if they are in opposite directions and have the same magnitude.
Therefore, the assertion and the reason both are the right and the reason is the right clarification of the assertion.
Therefore, the correct option is A.




Note:
The first point is to keep in mind that if the vectors are in opposite directions and lie on the same axis, then the angle between them will be \[{180^ \circ }\].