# How do you find the vector equation and the parametric equations of the line that passes through the points A (3, 4) and B (5, 5)?

Jan 26, 2017

vector eqn: $\vec{r} = \left(\begin{matrix}3 \\ 4\end{matrix}\right) + \lambda \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

parametric eqns:

$x = 3 + 2 \lambda$

$y = 4 + \lambda$

#### Explanation:

assuming we are working in 2D only

vector eqn of line :$\vec{r} = \vec{a} + \lambda \vec{d}$

$\vec{r} =$general point on the line

$\vec{a} =$known point on the line

$\vec{d} =$direction of line

$\lambda =$scalar

$\vec{d} = \vec{A B}$

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

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

vec(OA)=((3),(4)); vec(OB)=((5),(5))

$\vec{A B} = \vec{d} = = - \left(\begin{matrix}3 \\ 4\end{matrix}\right) + \left(\begin{matrix}5 \\ 5\end{matrix}\right) = \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

we can use either $\vec{O A} ,$or $\vec{O B}$ for $\vec{a}$

$\vec{r} = \left(\begin{matrix}3 \\ 4\end{matrix}\right) + \lambda \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

for the parametric eqns
let $\vec{r} = \left(\begin{matrix}x \\ y\end{matrix}\right)$

so $\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}3 \\ 4\end{matrix}\right) + \lambda \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

$x = 3 + 2 \lambda$

$y = 4 + \lambda$