Question: The value of $m$ for which the points with position vectors $\vec{a} = -\hat{i} - \hat{j} + 2\hat{k}$, $\vec{b} = 2\hat{i} + m\hat{j} + 5\hat{k}$ and $\vec{c} = 3\hat{i} + 11\hat{j} + 6\hat{k}$ are collinear, is:

Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]

Three points with position vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are collinear if and only if the vectors formed by them, $\vec{AB}$ and $\vec{BC}$, are parallel. This implies that $\vec{AB} = \lambda \vec{BC}$ for some scalar $\lambda$, or equivalently, their components must be proportional:

Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]

• Calculate vector $\vec{AB}$: $$\vec{AB} = (2 - (-1))\hat{i} + (m - (-1))\hat{j} + (5 - 2)\hat{k} = 3\hat{i} + (m+1)\hat{j} + 3\hat{k}$$
• Calculate vector $\vec{BC}$: $$\vec{BC} = (3 - 2)\hat{i} + (11 - m)\hat{j} + (6 - 5)\hat{k} = 1\hat{i} + (11-m)\hat{j} + 1\hat{k}$$
• Since the points are collinear, $\vec{AB} = k \vec{BC}$: $$3\hat{i} + (m+1)\hat{j} + 3\hat{k} = k(1\hat{i} + (11-m)\hat{j} + 1\hat{k})$$
• Comparing the $\hat{i}$ components: $3 = k(1) \implies k = 3$.
• Comparing the $\hat{j}$ components using $k=3$: $$m + 1 = 3(11 - m)$$ $$m + 1 = 33 - 3m$$ $$4m = 32 \implies m = 8$$

Step 3 (Final Concluded Answer): [1/3 Mark]

The value of $m$ is 8. Therefore, the correct option is (A).

💡 EL10 Concept Insight: Imagine three points are like beads on a single straight string; if you move from the first bead to the second, and then from the second to the third, you must be moving in the exact same direction. By checking if the "steps" (vectors) between these beads are just scaled-up versions of each other, we can prove they all lie on the same line.