Question

It has been given that $\vec{P}+\vec{Q}+\vec{R}=0$. Which of the following statements given is correct?
$\begin{align}
  & A.\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|=\left| {\vec{R}} \right| \\
 & B.\left| \vec{P}+\vec{Q} \right|=\left| {\vec{R}} \right| \\
 & C.\left| {\vec{P}} \right|-\left| {\vec{Q}} \right|=\left| {\vec{R}} \right| \\
 & D.\left| \vec{P}-\vec{Q} \right|=\left| {\vec{R}} \right| \\
\end{align}$

Answer 1

Hint: The vector addition and vector subtraction is the operation we have to perform while solving this question. Lots of mathematical operations can be conducted with and using the vectors. One of such operations has been mentioned as the addition of vectors. Two vectors can be added together in order to find out the resultant. This will help you in answering this question.

Complete step-by-step answer:
The 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 three vectors has been mentioned in the question. They are given as,
$\vec{P}+\vec{Q}+\vec{R}=0$
If this is zero, it means that the sum of the magnitudes of each of the vectors will be equivalent to zero. This can be written as,
\[\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|+\left| {\vec{R}} \right|=0\]
Let us check each of the equations using this result. As they are equal to zero, the first option will be wrong. When we do the relation in the first option we will obtain twice the value of \[\vec{R}\].
\[\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|+\left| {\vec{R}} \right|=\left| {\vec{R}} \right|+\left| {\vec{R}} \right|=2\left| {\vec{R}} \right|\]
Now when we check the second option, we can see that this relation can be true. That is,
\[\left| \vec{P}+\vec{Q} \right|=\left| {\vec{R}} \right|\]
The third and fourth options are also not giving the answer as zero. Therefore the answer for the question has been found to be equal to the option B.

So, the correct answer is “Option B”.

Note: A vector is defined as a body which is having both a magnitude and a direction. Schematically we can represent a vector as a line segment which is directed. The vector is having 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.