Answer
$${\bf{v}} = - \frac{{\sqrt 2 }}{4}{\bf{i}} - \frac{{\sqrt 2 }}{4}{\bf{j}}$$
Work Step by Step
$$\eqalign{
& \left\| {\bf{v}} \right\| = \frac{1}{2},{\text{ }}\theta = 225^\circ \cr
& {\text{The components of }}{\bf{v}}{\text{ are given by }} \cr
& {\bf{v}} = \left\| {\bf{v}} \right\|\left( {\cos \theta {\bf{i}} + \sin \theta {\bf{j}}} \right) \cr
& {\text{Then,}} \cr
& {\bf{v}} = \frac{1}{2}\left( {\cos 225^\circ {\bf{i}} + \sin 225^\circ {\bf{j}}} \right) \cr
& {\text{Simplifying}} \cr
& {\bf{v}} = \frac{1}{2}\left( { - \frac{{\sqrt 2 }}{2}{\bf{i}} - \frac{{\sqrt 2 }}{2}{\bf{j}}} \right) \cr
& {\bf{v}} = - \frac{{\sqrt 2 }}{4}{\bf{i}} - \frac{{\sqrt 2 }}{4}{\bf{j}} \cr} $$
