Question

If \[\vec{P}+\vec{Q}=\vec{R}\] and if \[\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|=\left| {\vec{R}} \right|\], then what is the angle between \[\vec{P}\] and \[\vec{Q}\]?

Answer 1

Hint: In the above question, the sum of two vectors is given and we have to find the angle between the two vectors. If two vectors are parallel to each other then the angle between them will be zero degrees. Two given vectors will be the same if they have the same magnitude and direction.

Complete step-by-step answer:
Vector tells us about the magnitude and direction of the given quantity. Vector is represented as the line segment and the length of the line segment gives us the magnitude of the vector and the arrow on one side of the line segment gives us the direction of the vector. All the quantities like force, velocity, and displacement are known as the vector quantities. The formula to find the magnitude of a vector gives us the length of the vector.
When a vector quantity is multiplied by a scalar quantity then their magnitude changes but their direction remains the same. If one vector is positive and the other one is negative then both the vectors will be in opposite directions to each other. One scalar quantity can divide another scalar quantity but one vector quantity cannot divide another vector quantity.
If two vectors are parallel to each other then their cross product will be zero.
In the above question, it is given that \[\vec{P}+\vec{Q}=\vec{R}\] and it is also given that \[\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|=\left| {\vec{R}} \right|\] and we have to find the angle between \[\vec{P}\] and \[\vec{Q}\].
We know that the magnitude sum of two vectors \[\vec{P}\] and \[\vec{P}\] is given as shown below.
\[{{\left( \left| {\vec{P}} \right|+\left| {\vec{Q}} \right| \right)}^{2}}=\left| {\vec{P}} \right|+\left| {\vec{Q}} \right|+2\left| {\vec{P}} \right|\left| {\vec{Q}} \right|\cos \theta \]
\[\vec{R}={{\left| {\vec{P}} \right|}^{2}}+{{\left| {\vec{Q}} \right|}^{2}}+2\left| {\vec{P}} \right|\left| {\vec{Q}} \right|\cos \theta \]……..eq(1)
Where the angle between the two vectors is \[\theta \].
In the question above it is also given that.
\[\begin{align}
  & \left( \vec{P}+\vec{Q} \right)=\vec{R} \\
 & \Rightarrow \vec{R}={{(\vec{P})}^{2}}+{{(\vec{Q})}^{2}}+2\vec{P}\vec{Q} \\
\end{align}\]
……eq(2)
Eq(1) and eq(2) are equal. So the result is as follows.
\[{{\left| {\vec{P}} \right|}^{2}}+{{\left| {\vec{Q}} \right|}^{2}}+2\left| {\vec{P}} \right|\left| {\vec{Q}} \right|\cos \theta ={{(\vec{P})}^{2}}+{{(\vec{Q})}^{2}}+2\vec{P}\vec{Q}\]
We know that,
\[\left| {\vec{P}} \right|=\vec{P}\]
\[\left| {\vec{Q}} \right|=\vec{Q}\]
So the result will be as shown below.
\[2\left| {\vec{P}} \right|\left| {\vec{Q}} \right|\cos \theta =2\vec{P}\vec{Q}\]
\[\Rightarrow \cos \theta =1\].
So \[\cos \theta \] comes out to be when the angle between both the vectors comes out to zero.
So the angle between the vector \[\vec{P}\] and \[\vec{Q}\] is \[0{}^\circ \]
So, the correct answer is “\[0{}^\circ \]”.

Note: There are two types of quantities- scalar quantities and vector quantities. Scalar quantities are those quantities that have magnitude only and no direction and vector quantities are those quantities that have magnitude and direction both. The magnitude of the sum of two vectors will be equal to the magnitude of the difference between the two vectors.