Answer
a) The desired vector is $\text{v}=\text{i +2j}$
b) The value of $\left\| \text{v} \right\|$ is $\sqrt{5}$.
Work Step by Step
(a)
Let us represent the given two points as:
$\begin{align}
& \left( {{x}_{1}},{{y}_{1}} \right)=\left( -2,3 \right)\, \\
& \,\left( {{x}_{2}},{{y}_{2}} \right)=\left( -1,5 \right)
\end{align}$.
A position vector between two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ denoted in vector form is:
$\mathbf{v}\text{ = }\left( {{x}_{2}}-{{x}_{1}} \right)\mathbf{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\mathbf{j}$ …… (1)
Now substitute the value of $\left( {{x}_{1}},{{y}_{1}} \right)\,\,\text{ and }\,\,\left( {{x}_{2}},{{y}_{2}} \right)$ in equation (1),
$\begin{align}
& \mathbf{v}\text{ = }\left( -1+2 \right)\mathbf{i}+\left( 5-3 \right)\mathbf{j} \\
& \text{= }\mathbf{i}+\text{2}\mathbf{j}
\end{align}$
(b)
The vector v can be computed as:
$\mathbf{v}=\mathbf{i}+2\mathbf{j}$
Here, $\left\| \mathbf{v} \right\|$ represents the magnitude of the vector, which is denoted by the formula:
$\left\| \mathbf{v} \right\|=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Now in $\mathbf{v}=\mathbf{i}+2\mathbf{j}$ ;
$\begin{align}
& x=1\, \\
& y=2
\end{align}$
Then,
$\begin{align}
& \left\| \text{v} \right\|=\sqrt{{{1}^{2}}+{{2}^{2}}} \\
& =\sqrt{5}
\end{align}$
