Answer
a) $\frac{4}{5} \hat{\jmath}-\frac{3}{5} \hat{\imath}$
b) $-\frac{1}{3} \hat{\jmath}-\frac{2}{3} \hat{k}+\frac{2}{3} \hat{\imath}$
c) $-\frac{3}{5} \hat{\jmath}+\frac{4}{5} \hat{\imath}$
Work Step by Step
The unitary vector in the direction of a given vector $\vec{v}$ is
\[
\vec{v} \frac{1}{\|\vec{v}\|}=\hat{v}
\]
a) The unitary vector in the opposite direction of $-4 \hat{\jmath}+3 \hat{\imath}=\vec{v}$ is
\[
-\hat{v}=- \vec{v} \frac{1}{\|\vec{v}\|}=-\frac{1}{\sqrt{(-4)^{2}+3^{2}}}(3 \hat{\imath}-4 \hat{\jmath})=\frac{4}{5} \hat{\jmath}-\frac{3}{5} \hat{\imath}
\]
b) The unitary vector in the same direction of $\vec{v}=-\hat{\jmath}-2 \hat{k}+2 \hat{\imath}$ is
\[
\begin{aligned}
\hat{v} &=\frac{1}{\|\vec{v}\|} \vec{v}=\frac{1}{\sqrt{2^{2}+(-1)^{2}+(-2)^{2}}}(2 \hat{\imath}-\hat{\jmath}-2 \hat{k}) \\
&=-\frac{1}{3} \hat{\jmath}-\frac{2}{3} \hat{k}+\frac{2}{3} \hat{\imath}
\end{aligned}
\]
$c$
Notice that
\[
\vec{v}=\overrightarrow{A B}=(-1-2) \hat{\jmath}+(1-(-3)) \hat{\imath}=-3 \hat{\jmath}+4 \hat{\imath}
\]
and then
\[
\hat{v}=\frac{1}{\|\vec{v}\|} \vec{v}=(4 \hat{\imath}-3 \hat{\jmath}) \frac{1}{\sqrt{(-3)^{2}+4^{2}}}=-\frac{3}{5} \hat{\jmath}+\frac{4}{5} \hat{\imath}
\]
