Answer
$a)$
\[
B(4,-4)=B(c, d)
\]
the endpoint.
$\mathrm{b})$
\[
A(8,-1,-3)
\]
is the initial point.
Work Step by Step
a) Let $\vec{v}$ be a vector in 2 -space with initial point $A(a, b)$ and point $B(c, d),$; then
\[
\vec{v}=\overrightarrow{A B}=\langle c-a, d-b\rangle=(d-b) \hat{\jmath}+(c-a) \hat{\imath}
\]
It's given that $\vec{v}=-2 \hat{\jmath}+3 \hat{\imath},$ so:
\[
-2=-b+d \text { and } 3=-a+c
\]
We also get:
\[
A(a, b)=A(1,-2) \Rightarrow b=-2 \text { and } a=1
\]
and then
\[
4=c=a+3 \quad \text { and } \quad -4=-2+b=d
\]
So, we have that
\[
B(4,-4)=B(c, d)
\]
is the terminal point of the vector.
b) Let $A(a, b, c)$ the initial point, and then
\[
\vec{v}=\overrightarrow{A B}
\]
where $B(5,0,-1)$ is the terminal point. So
\[
\begin{array}{l}
\langle-3,1,2\rangle=\langle 5-a, 0-b,-1-c\rangle \\
\quad -3=-a+5 \\
1=-b+0 \\
2=-c-1
\end{array} \Rightarrow\left\{\begin{array}{l}
b=-1 \\
c=-3 \\
a=8
\end{array}\right.
\]
So $A(8,-1,-3)$ is the initial point
