# What is the equation of the line passing through (48,7) and (93,84)?

Jan 8, 2016

The common method is to use the determinant

$A \left(48 , 7\right)$ $B \left(93 , 84\right)$

The vector formed by $A$ and $B$ is :
$\vec{A B} = \left(93 - 48 , 84 - 7\right) = \left(45 , 77\right)$

(which is a vector director to our line)

and now imagine a point $M \left(x , y\right)$ it can be anything

the vector formed by $A$ and $M$ is ;
$\vec{A M} = \left(x - 48 , y - 7\right)$

$\vec{A B}$ and $\vec{A M}$ are parallel if and only if $\det \left(\vec{A B} , \vec{A M}\right) = 0$

in fact they will be parallel and be on the same line, because they share the same point $A$

Why if $\det \left(\vec{A B} , \vec{A M}\right) = 0$ they are parallel ?

because $\det \left(\vec{A B} , \vec{A M}\right) = A B \cdot A M \sin \left(\theta\right)$ where $\theta$ is the angle formed by the two vectors, since the vectors are not $= \vec{0}$ the only way $\det \left(\vec{A B} , \vec{A M}\right) = 0$ it is $\sin \left(\theta\right) = 0$

and $\sin \left(\theta\right) = 0$ when $\theta = \pi$ or $= 0$ if the angle between two line $= 0$ or $= \pi$ they are parallel (Euclide definition)

compute the $\det$ and find

$45 \left(y - 7\right) - 77 \left(x - 48\right) = 0$

And voilà ! You know how to do it geometrically ; )