Question

Given that \[\overrightarrow R = \overrightarrow P + \overrightarrow Q \]. Which of the following relations is necessarily valid?
A. \[P < Q\]
B. \[P > Q\]
C. \[P = Q\]
D. None of the above

Answer 1

Hint: The resultant vector is the sum of two vectors. The relation between the two vectors i.e which one is greater can be identified by assuming some values taken into them. With these values, the resultant vector has to be checked. Thereafter, the condition has to be checked and found the right option.

Complete step by step answer:
Vector addition is the process or operation of adding two or more vectors together. And vector subtraction is the process or operation of subtracting one vector from the other. Here we can see that the sum of two vectors has been mentioned in the question. They are given as,
\[\overrightarrow R = \overrightarrow P + \overrightarrow Q - - - - - - - \left( 1 \right)\]

For example, we assume \[\overrightarrow R = 2\].
Based on the first condition \[\left( {P < Q} \right)\], let us assume\[P = 1,Q = 2\]. Substitute these values in equation (1). We get,
\[\overrightarrow R = \overrightarrow P + \overrightarrow Q = 1 + 2 = 3\]
We know that, \[\overrightarrow R = 2\]
\[2 \ne 3\]. So, the relation \[P < Q\] is not valid. Hence, option A is incorrect.

Based on the second condition \[\left( {P > Q} \right)\], let us assume\[P = 2,Q = 1\]. Substitute these values in equation 1. We get,
\[\overrightarrow R = \overrightarrow P + \overrightarrow Q = 2 + 1 = 3\]
We know that, \[\overrightarrow R = 2\]
\[2 \ne 3\]. So, the relation \[P > Q\] is not valid. Hence, option B is incorrect.

Based on the third condition \[\left( {P = Q} \right)\], Let us assume\[P = 1,Q = 1\]. Substitute these values in equation 1. We get,
\[\overrightarrow R = \overrightarrow P + \overrightarrow Q = 1 + 1 = 2\]
We know that, \[\overrightarrow R = 2\]
\[2 = 2\]. So, the relationship \[P = Q\] is necessarily valid.

Hence, option C is correct.

Note: A vector is termed as a quantity which has both a magnitude and a direction. Schematically we will represent a vector as a line segment that is directed. The vector has the length as the magnitude of the vector and with an arrow representing the direction. The direction of the vector is basically from its tail to its head.