And another
Certainly they are not parallel, so they must intersect. WLOG assume m1 > m2.
We obtain the intersection point via:
m1x+c1=y=m2x+c2⟹x=m1−m2c2−c1
⟹y=m1m1−m2c2−c1+c1=m2m1−m2c2−c1+c2
This is not necessary, but we should know
it exists.
We may form two vectors along the lines like so (x and y exist as the above quantities):
P1=(x,y)=Q1,P2=(x+1,y+m1),Q2=(x+1,y+m2)
(You can verify the above by pluggin in (x+1) in for x and seeing it comes out to y+slope)
Notice
Pˉ is a vector lying on the first line,
Qˉ on the second. To verify, you may take any point on the vector within the domain (x,x+1) and show it satisfies the linear equation.
Pˉ=<1,m1>
Qˉ=<1,m2>
Taking the dot product we see:
Pˉ⋅Qˉ=1+m1m2=1+(−1)=0
Hene vectors
Pˉ and
Qˉ are orthogonal, and so are the lines.