Answer
(a) The vector $ - {\bf{w}}$ matches (ii) $\overrightarrow {CA} $
(b) The vector $ - {\bf{v}}$ matches (iv) $\overrightarrow {BA} $
(c) The vector ${\bf{w}} - {\bf{v}}$ matches (iii) $\overrightarrow {BC} $
(d) The vector ${\bf{v}} - {\bf{w}}$ matches (i) $\overrightarrow {CB} $
Work Step by Step
(a) The vector $ - {\bf{w}}$ has the same length with the vector ${\bf{w}} = \overrightarrow {AC} $ but in opposite direction. So, it matches (ii) $\overrightarrow {CA} $.
(b) The vector $ - {\bf{v}}$ has the same length with the vector ${\bf{v}} = \overrightarrow {AB} $ but in opposite direction. So, it matches (iv) $\overrightarrow {BA} $.
(c) Write ${\bf{w}} - {\bf{v}} = {\bf{w}} + \left( { - {\bf{v}}} \right)$.
From the result in part (b) we have $ - {\bf{v}} = \overrightarrow {BA} $. So,
${\bf{w}} - {\bf{v}} = {\bf{w}} + \left( { - {\bf{v}}} \right) = \overrightarrow {AC} + \overrightarrow {BA} = \overrightarrow {BA} + \overrightarrow {AC} $.
${\bf{w}} - {\bf{v}} = \overrightarrow {BC} $
It matches (iii) $\overrightarrow {BC} $.
(d) Write ${\bf{v}} - {\bf{w}} = {\bf{v}} + \left( { - {\bf{w}}} \right)$.
From the result in part (a) we have $ - {\bf{w}} = \overrightarrow {CA} $. So,
${\bf{v}} - {\bf{w}} = {\bf{v}} + \left( { - {\bf{w}}} \right) = \overrightarrow {AB} + \overrightarrow {CA} = \overrightarrow {CA} + \overrightarrow {AB} $.
${\bf{v}} - {\bf{w}} = \overrightarrow {CB} $.
It matches (i) $\overrightarrow {CB} $.
