A vector has a magnitude x. If it is rotated | vedclass

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Question

(A) The magnitude of the change in a vector $\vec{A}$ of magnitude $x$ when rotated by an angle $	heta$ is given by $|\Delta \vec{A}| = |\vec{A}' - \

Vedclass · Accepted Answer

$(A \rightarrow q, B \rightarrow r, C \rightarrow p, D \rightarrow s)$

Vedclass · Answer

(A) The magnitude of the change in a vector $\vec{A}$ of magnitude $x$ when rotated by an angle $	heta$ is given by $|\Delta \vec{A}| = |\vec{A}' - \vec{A}| = 2x \sin(	heta/2)$.Given $|\Delta \vec{A}| = nx$, we have $nx = 2x \sin(	heta/2)$, which implies $n = 2 \sin(	heta/2)$.For $(A)$ $	heta=60^{\circ}$: $n = 2 \sin(30^{\circ}) = 2(1/2) = 1$. Thus, $(A ightarrow q)$.For $(B)$ $	heta=90^{\circ}$: $n = 2 \sin(45^{\circ}) = 2(1/\sqrt{2}) = \sqrt{2}$. Thus, $(B ightarrow r)$.For $(C)$ $	heta=120^{\circ}$: $n = 2 \sin(60^{\circ}) = 2(\sqrt{3}/2) = \sqrt{3}$. Thus, $(C ightarrow p)$.For $(D)$ $	heta=180^{\circ}$: $n = 2 \sin(90^{\circ}) = 2(1) = 2$. Thus, $(D ightarrow s)$.Therefore, the correct matching is $(A ightarrow q, B ightarrow r, C ightarrow p, D ightarrow s)$.

Column $I$	Column $II$
$(A)$ $\theta=60^{\circ}$	$(p)$ $n=\sqrt{3}$
$(B)$ $\theta=90^{\circ}$	$(q)$ $n=1$
$(C)$ $\theta=120^{\circ}$	$(r)$ $n=\sqrt{2}$
$(D)$ $\theta=180^{\circ}$	$(s)$ $n=2$

Column $I$	Column $II$
$(A)$ Component along $x$-axis	$(p)$ $5 \text{ unit}$
$(B)$ Component along vector $(2 \hat{i} + \hat{j} + 2 \hat{k})$	$(q)$ $4 \text{ unit}$
$(C)$ Component along $(6 \hat{i} + 8 \hat{j} - 10 \hat{k})$	$(r)$ $0$
$(D)$ Component along $(-3 \hat{i} - 4 \hat{j} + 5 \hat{k})$	$(s)$ None

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