Passage: $P(-2,-3,5)$, $Q(4,-1,5)$, $R(6,-4,8)$ and $S(2,-6,10)$ are four points.

Question: Consider the following statements :
I. $PQ$ is parallel to $RS$.
II. $PR$ is perpendicular to $QS$.
Which of the statements given above is/are correct ?

The direction ratios of the line segment $PQ$ are $4 - (-2), -1 - (-3), 5 - 5$, which gives $\langle 6, 2, 0 \rangle$.

The direction ratios of the line segment $RS$ are $2 - 6, -6 - (-4), 10 - 8$, which gives $\langle -4, -2, 2 \rangle$.

Since the ratios of their corresponding direction ratios are not equal ($\dfrac{6}{-4} \neq \dfrac{2}{-2} \neq \dfrac{0}{2}$), $PQ$ is not parallel to $RS$. Thus, Statement I is incorrect.

The direction ratios of the line segment $PR$ are $6 - (-2), -4 - (-3), 8 - 5$, which gives $\langle 8, -1, 3 \rangle$.

The direction ratios of the line segment $QS$ are $2 - 4, -6 - (-1), 10 - 5$, which gives $\langle -2, -5, 5 \rangle$.

For $PR$ to be perpendicular to $QS$, the dot product of their direction vectors must be zero. Calculating the dot product:

$\vec{PR} \cdot \vec{QS} = (8)(-2) + (-1)(-5) + (3)(5) = -16 + 5 + 15 = 4$

Since the dot product is not zero, $PR$ is not perpendicular to $QS$. Thus, Statement II is incorrect.

Therefore, neither Statement I nor Statement II is correct.

Answer: Neither I nor II