Question

Let the angle between two non-zero vectors $\overrightarrow{A}$and $\overrightarrow{B}$ be ${{120}^{\circ }}$and its resultant be $\overrightarrow{C}$
(A) $C$ must be equal to $|\overrightarrow{A}-\overrightarrow{B}|$
(B) $C$ must be less than $|\overrightarrow{A}-\overrightarrow{B}|$
(C) $C$ must be greater than $|\overrightarrow{A}-\overrightarrow{B}|$
(D) $C$ may be equal to $|\overrightarrow{A}-\overrightarrow{B}|$

Answer 1

Hint:We are given two vectors and the angle between them. When these values are given, we can easily find the resultant by parallelogram law. We’re given some conditions related to the resultant. They are to be compared with the calculated resultant.

Formulas used:
Parallelogram law:
If A and B are two vectors and the angle between them is $\theta $
Then the magnitude of the resultant $|\overrightarrow{R}|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }$

Complete step by step solution:
We are given two non-zero vectors $\overrightarrow{A}$and $\overrightarrow{B}$ and the angle between them is${{120}^{\circ }}$. Its resultant is $\overrightarrow{C}$. By parallelogram law, we know that
$|\overrightarrow{R}|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }$
Substituting values, we get
\[\begin{align}
& |\overrightarrow{C}|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos 120} \\
&\Rightarrow|\overrightarrow{C}| =\sqrt{{{A}^{2}}+{{B}^{2}}-AB} \\
\end{align}\] …………… (1)
When we extend $\overrightarrow{B}$ to the opposite direction, the angle formed between $\overrightarrow{A}$and $\overrightarrow{B}$ will be ${{60}^{\circ }}$
Hence,
$\begin{align}
& |\overrightarrow{A}-\overrightarrow{B}|=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos {{60}^{\circ }}} \\
&\Rightarrow |\overrightarrow{A}-\overrightarrow{B}|=\sqrt{{{A}^{2}}+{{B}^{2}}+AB} \\
\end{align}$…………………….(2)
Upon comparing (1) and (2), we can say that the magnitude of the resultant is less than the magnitude of the difference of the component vectors that is
$\overrightarrow{C}<|\overrightarrow{A}-\overrightarrow{B}|$

Hence we can say that option B is the correct answer among the given options.

Note:We can include additional values into the resultant value that help in the simplification of the root and then compare it with the given conditions in options. We can convert the values inside the root to mathematical identities and avoid the root, find out the change which is made to do the conversion, and conclude whether it satisfies the condition.