proof

red and white kop!

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two lines have equations y=m(1)x + c(1) and y = m(2)x + c(2), and m(1)m(2) = -1, proove that the lines are perpendicular.
im aware that this is supposed to be an extension question but still i dont know what to use in order to proove this, seeing that m(1)m(2) = -1 is apparently not enough. can somebody help with a clear proof on how to do this?
 
Here's another.

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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=c2c1m1m2\displaystyle m_1x + c_1 = y = m_2x + c_2 \,\, \implies \,\, x = \frac{c_2-c_1}{m_1-m_2}

        y=m1c2c1m1m2+c1=m2c2c1m1m2+c2\displaystyle \,\, \implies \,\, y = m_1 \frac{c_2-c_1}{m_1-m_2} +c1 = m_2 \frac{c_2-c_1}{m_1-m_2} +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)\displaystyle P_1 = (x,y) = Q_1, P_2 = (x+1,y+m_1), Q_2 = (x+1,y+m_2)

(You can verify the above by pluggin in (x+1) in for x and seeing it comes out to y+slope)

Notice Pˉ\displaystyle \bar{P} is a vector lying on the first line, Qˉ\displaystyle \bar{Q} 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>\displaystyle \bar{P} = <1, m_1>
Qˉ=<1,m2>\displaystyle \bar{Q} =<1, m_2>

Taking the dot product we see:

PˉQˉ=1+m1m2=1+(1)=0\displaystyle \bar{P} \cdot \bar{Q} = 1 + m_1m_2 = 1 + (-1) = 0

Hene vectors Pˉ\displaystyle \bar{P} and Qˉ\displaystyle \bar{Q} are orthogonal, and so are the lines.
 
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