The two lines $x=ay+b,z=cy+d$; and $x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$ are perpendicular to each other if:

Rewrite the equations of the lines in symmetric form:Line 1: $\frac{x-b}{a}=y=\frac{z-d}{c}$ The direction ratios are $(a,1,c)$.Line 2: $\frac{x-b^{\prime }}{a^{\prime }}=y=\frac{z-d^{\prime }}{c^{\prime }}$ The direction ratios are $(a^{\prime },1,c^{\prime })$.The lines are perpendicular if the dot product of their direction vectors is zero:$a\cdot a^{\prime }+1\cdot 1+c\cdot c^{\prime }=0$$a{a}^{\prime }+c{c}^{\prime }+1=0⟹a{a}^{\prime }+c{c}^{\prime }=-1$.