# What is the slope of any line perpendicular to the line passing through (3,12) and (-5,17)?

Dec 31, 2015

Of any line ?

$A = \left(3 , 12\right)$ $B = \left(- 5 , 17\right)$

$\vec{A B} = \left(- 5 - 3 , 17 - 12\right) = \left(- 8 , 5\right)$

The equation of the line directed by this vector is $P = 5 x + 8 y = 0$

Now imagine all the couple which are solutions to this equation

lambda = (x_0,x_1,...x_n;y_0,y_1,...y_n)

Note that $A , B \in \lambda$

Now imagine an arbitrary coordinate $M \left(x , y\right)$ It can be anything

$\vec{\lambda M}$ is perpendicular to $P$ if and only if it is perpendicular to $\vec{A B}$ and it is perpendicular to $\vec{A B}$ if and only if $\vec{\lambda M} \cdot \vec{A B} = 0$

$- 8 \left(x - {x}_{0}\right) + 5 \left(y - {y}_{0}\right) = 0$ if you take the point $A$ you have

$- 8 \left(x - 3\right) + 5 \left(y - 12\right) = 0$

if you take the point $B$ you have :

$- 8 \left(x + 5\right) + 5 \left(y - 17\right) = 0$

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