## Calculus (3rd Edition)

(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) $\overrightarrow {CB}$
(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) $\overrightarrow {CB}$.