If the vectors \( \vec{a} = 3\hat{i} - p\hat{j} + 5\hat{k} \) and \( \vec{b} = -6\hat{i} + 14\hat{j} + q\hat{k} \) are collinear, then the value of \( p \) and \( q \) are:

\( p = 7, q = -10 \)

The correct answer is Option (3) → \( p = 7, q = -10 \)

Given vectors: $\vec{a} = 3\hat{i} - p\hat{j} + 5\hat{k}$ and $\vec{b} = -6\hat{i} + 14\hat{j} + q\hat{k}$

Condition: Vectors are collinear ⟺ $\vec{b} = \lambda \vec{a}$ for some scalar $\lambda$

Compare components:

$-6 = \lambda \cdot 3 \Rightarrow \lambda = \frac{-6}{3} = -2$

Use $\lambda = -2$ in other components:

$14 = \lambda \cdot (-p) = -2(-p) = 2p \Rightarrow p = \frac{14}{2} = 7$

$q = \lambda \cdot 5 = -2 \cdot 5 = -10$

$p = 7,\ q = -10$