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Given below in Column $-I$ are the relations between vectors $\vec a \,$ $\vec b \,$ and $\vec c \,$ and in Column $-II$ are the orientations of $\vec a$, $\vec b$ and $\vec c$ in the $XY-$ plane. Match the relation in Column $-I$ to correct orientations in Column $-II$.
| Column $-I$ | Column $-II$ |
| $(a)$ $\vec a \, + \,\,\vec b \, = \,\,\vec c $ | $(i)$ Image |
| $(b)$ $\vec a \, - \,\,\vec c \, = \,\,\vec b$ | $(ii)$ Image |
| $(c)$ $\vec b \, - \,\,\vec a \, = \,\,\vec c $ | $(iii)$ Image |
| $(d)$ $\vec a \, + \,\,\vec b \, + \,\,\vec c =0$ | $(iv)$ Image |
Solution
Consider the below given diagram in which vectors $A$ and $B$ are connected by head and tail.
Resultant vector $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$
$(a)$ from $(iv)$ it is clear that $\mathrm{c}=\mathrm{a}+\mathrm{b}$
$(iv)$ matches with $(a)$
$(b)$ from (iii) $\mathrm{c}+\mathrm{b}=\mathrm{a}$
$ \Rightarrow \mathrm{a}-\mathrm{c}=\mathrm{b}$
$(iii)$ matches with $(b)$
$(c)$ from $(i)$ $\mathrm{b}=\mathrm{a}+\mathrm{c} \Rightarrow \mathrm{b}-\mathrm{a}=\mathrm{c}$
$(ii)$ matches with $(d)$
$(d)$ from (ii) $-\mathrm{c}=\mathrm{a}+\mathrm{b} \Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=0$
$(i)$ matches with $(c)$
