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
$R=\sqrt{A^{2}+B^{2}+2 A B \cos \theta}$
$\therefore$ 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}-2 a^{2} \cos \left(180^{\circ}\right)} $
$=\sqrt{2 a^{2}-2 a^{2}} \left[\because \cos 180^{\circ}=-1\right]$
$\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.