Question

Assertion Magnitude of resultant of two vectors may be less than the magnitude of either vector.
Reason Vector addition is commutative.

• A. Both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
• B. Both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
• C. Assertion is correct but Reason is incorrect.
• D. Assertion is incorrect but Reason is correct.

Answer 1

Answer: (B)

Resultant of two vectors $A$ and $B$ is given as
$R=\sqrt{A^2+B^2+2AB\cos \theta }$
$\therefore$ We can say that
(i) If $\theta$ is an obtuse angle, then magnitude of $R$ will be less than magnitude of the either vectors $A$ or $B$.
e.g. if $|A|=4,|B|=3$ and $\theta =120^{\circ }$, then
$|R|=\sqrt{4^2+3^2+2\times 4\times 3\cos \left(120^{\circ }\right)}$
$=\sqrt{25-12}=\sqrt{13}\left(\because \cos 120^{\circ }=-\frac{1}{2}\right)$
$\therefore |R|<|A|$
(ii) If the vectors are in opposite direction and are equal in magnitude, then also the magnitude of $R$ will be less than the magnitude of either vectors $A$ or $B$.
e.g., if $|A|=|B|=a$ (say) and $\theta =180^{\circ }$
then,
$|R|=\sqrt{a^2+a^2-2a^2\cos \left(180^{\circ }\right)}$
$=\sqrt{2a^2-2a^2}\left[\because \cos 180^{\circ }=-1\right]$
$\therefore |R|<|A|$ or $|B|$
Also, vector addition is commutative in nature.
$A+B=B+A$
Therefore, Assertion and Reason are correct but Reason is not the correct explanation of Assertion.