Answer
(a) $\vec {AC}$
(b) $\vec {CB}$
(c) $\vec {DA}$
(d) $\vec {DB}$
Work Step by Step
(a) The triangle rule states that if A,B and C form a triangle then $ \vec {AB}+\vec {BC}=\vec {AC}$.
In this case A,B and C form a triangle . Moving from A to B and then from B to C produces the same result as moving directly from A to C.
(b) The triangle rule states that if A,B and C form a triangle then $ \vec {AB}+\vec {BC}=\vec {AC}$.
In this case C,D and B form a triangle . Moving from C to D and then from D to B produces the same result as moving directly from C to B.
Thus, $ \vec {CD}+\vec {DB}=\vec {CB}$
(c) The triangle rule states that if A,B and C form a triangle then $ \vec {AB}+\vec {BC}=\vec {AC}$.
As we know: $\vec {AB}=-\vec {BA}$ (basic vector algebra)
Thus, $ \vec {DB}-\vec {AB}=\vec {DB}+\vec {BA}=\vec {DA}$
(d) Add the first two vectors to get the result $\vec {DA}$
$ \vec {DC}+\vec {CA}=\vec {DA}$
Add the third vector to get the result $\vec {DB}$
$ \vec {DA}+\vec {AB}=\vec {DB}$
This is equivalent to $(\vec {DC}+\vec {CA})+\vec {AB}=\vec {DB}$
