# Given the initial (-3,1) and terminal (4,5) points of a vector, how do you write a linear combination of the standard unit vector i and j?

Aug 22, 2017

$\boldsymbol{\underline{v}} = \left(\begin{matrix}7 \\ 4\end{matrix}\right) = \left\langle7 , 4\right\rangle = 7 \boldsymbol{\underline{\hat{i}}} + 4 \boldsymbol{\underline{\hat{j}}}$

#### Explanation:

Denote the given coordinates by:

$O = \left(0 , 0\right)$
$A = \left(- 3 , 1\right)$
$B = \left(4 , 5\right)$

Then the vector $\boldsymbol{\underline{v}}$ is given by:

$\boldsymbol{\underline{v}} = \boldsymbol{\vec{O B}} - \boldsymbol{\vec{O A}}$
$\setminus \setminus \setminus = \left(\begin{matrix}4 \\ 5\end{matrix}\right) - \left(\begin{matrix}- 3 \\ 1\end{matrix}\right)$
$\setminus \setminus \setminus = \left(\begin{matrix}7 \\ 4\end{matrix}\right)$

Alternatively, depending upon the desired notation we can also write:

$\boldsymbol{\underline{v}} = \left\langle7 , 4\right\rangle = 7 \boldsymbol{\underline{\hat{i}}} + 4 \boldsymbol{\underline{\hat{j}}}$