# How do you find the unit vector that has the same direction as the vector from the point A= (5, 2) to the point B= (3, 2)?

Jan 7, 2017

$\left(\begin{matrix}- 1 \\ 0\end{matrix}\right) = - \hat{\vec{i}}$

#### Explanation:

if $A$ has coordinate $\left(5 , 2\right)$ then its position vector is given by

$\vec{O A} = 5 \hat{\vec{i}} + 2 \hat{\vec{j}} = \left(\begin{matrix}5 \\ 2\end{matrix}\right)$

if $B$ has coordinate $\left(3 , 2\right)$ then its position vector is given by

$\vec{O B} = 3 \hat{\vec{i}} + 2 \hat{\vec{j}} = \left(\begin{matrix}3 \\ 2\end{matrix}\right)$

The vector from $A$ to$B$ is:

$\vec{A B} = \vec{A O} + \vec{O B}$

$\vec{A B} = - \vec{O A} + \vec{O B}$

$\vec{A B} = - \left(\begin{matrix}5 \\ 2\end{matrix}\right) + \left(\begin{matrix}3 \\ 2\end{matrix}\right)$

$\vec{A B} = \left(\begin{matrix}- 5 + 3 \\ - 2 + 2\end{matrix}\right) = \left(\begin{matrix}- 2 \\ 0\end{matrix}\right)$

call this vector $\vec{d}$

A unit vector in this direction is given by

$\hat{\vec{d}} = \frac{\vec{d}}{|} \vec{d} |$

$\vec{d} = \left(\begin{matrix}- 2 \\ 0\end{matrix}\right) \implies | \vec{d} | = 2$

$\hat{\vec{d}} = \frac{1}{2} \left(\begin{matrix}- 2 \\ 0\end{matrix}\right) = \left(\begin{matrix}- 1 \\ 0\end{matrix}\right)$