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The given lines are $x = py + q$ and $z = ry + s$. Rearranging these, we get: $$ \frac{x - q}{p} = y = \frac{z - s}{r} $$ The direction vector of the first line is $\vec{b_1} = (p, 1, r)$.
Similarly, for the second line $x = p'y + q'$ and $z = r'y + s'$, we get: $$ \frac{x - q'}{p'} = y = \frac{z - s'}{r'} $$ The direction vector of the second line is $\vec{b_2} = (p', 1, r')$.
Two lines are perpendicular if the dot product of their direction vectors is zero: $$ \vec{b_1} \cdot \vec{b_2} = 0 $$ $$ (p)(p') + (1)(1) + (r)(r') = 0 $$ $$ pp' + 1 + rr' = 0 $$ $$ pp' + rr' = -1 $$
The statement claims the condition is $pp' + rr' = 1$, but the derivation shows the condition is $pp' + rr' = -1$. Therefore, the assertion is false.
Final Answer: The assertion is False.