How to prove that the product of gradient of two perpendicular lines is equal to -1?

If m1 and m2 are the gradients to two perpendicular lines, prove that m1*m2 = -1.How to prove that the product of gradient of two perpendicular lines is equal to -1?
say we have three points:





P1 (0,0)


P2 (x1,y1)


P3 (x2,y2)





line1 goes through P1 and P2


line 2 goes through P1 and P3





m1 = (y1-0)/(x1-0) = y1/x1


m2 = (y2-0)/(x2-0) = y2/x2





define each line as vectors:


vector1 = %26lt;x1,y1%26gt;


vector2 = %26lt;x2,y2%26gt;





these vectors are perpendicular if they're dot product equals zero


x1x2 + y1y2 = 0


x1x2 = -y1y2 rearrange


x1/y1 = -y2/x2


-x1/y1 = y2/x2





substitute into m formulas:


m1 = y1/x1


m2 = y2/x2 we know that y2/x2 = -x1/y1


m2 = -x1/y1





product the two slopes





m1m2 = (y1/x1)(-x1/y1) = -1





voila