Answer
$\therefore\vec{V_{AB}}$ = $\text{30+40 = 70 m/s}$
$\therefore \vec{V_{AB}}$ = $\text{50+(-20) = 30 m/s}$
$\therefore\vec{V_{AB}}$ = $\text{(-20)+60 = 40 m/s}$
$\therefore\vec{V_{AB}}$ = $\text{(-50)-10 = -60 m/s}$
Work Step by Step
$\vec{V_{AB}}$ = $\vec{V_{A}} - \vec{V_{B}}$
Similarly
$\vec{V_{AC}}$ = $\vec{V_{A}} - \vec{V_{C}}$
$\vec{V_{CB}}$ = $\vec{V_{C}} - \vec{V_{B}}$
To find:-
$\text{Case a)}$ $\vec{V_{AB}}$=?
But
$\vec{V_{AB}}$ = $\vec{V_{A}} - \vec{V_{C}} + \vec{V_{C}} - \vec{V_{B}}$
$\vec{V_{AB}}$ = $\vec{V_{AC}}+ \vec{V_{CB}}$
$\vec{V_{AB}}$ = $\text{30+40 = 70 m/s}$
Similarly,
$\text{Case b)}$ $\vec{V_{AB}}$=?
$\vec{V_{AB}}$ = $\vec{V_{A}} - \vec{V_{C}} + \vec{V_{C}} - \vec{V_{B}}$
$\vec{V_{AB}}$ = $\vec{V_{AC}}+ \vec{V_{CB}}$
$\vec{V_{AB}}$ = $\text{50+(-20) = 30 m/s}$
$\text{Case c)}$ $\vec{V_{CB}}$=?
Similarly
$\vec{V_{CB}}$ = $\vec{V_{C}} - \vec{V_{A}} + \vec{V_{A}} - \vec{V_{B}}$
$\vec{V_{AB}}$ = $-\vec{V_{AC}}+ \vec{V_{CB}}$
$\vec{V_{AB}}$ = $\text{(-20)+60 = 40 m/s}$
$\text{Case d)}$ $\vec{V_{AC}}$=?
Similarly
$\vec{V_{AC}}$ = $\vec{V_{A}} - \vec{V_{B}} + \vec{V_{B}} - \vec{V_{C}}$
$\vec{V_{AB}}$ = $\vec{V_{AB}} - \vec{V_{CB}}$
$\vec{V_{AB}}$ = $\text{(-50)-10 = -60 m/s}$