Question

Let $\vec{OA}=2\hat{i}+3\hat{j}-5\hat{k}$, $\vec{OB}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{OC}=6\hat{i}-5\hat{j}+7\hat{k}$ be position vectors of A, B and C. Then ________.

• A. $\vec{AC}=3\vec{AB}$
• B. $\vec{AB}=3\vec{BC}$
• C. $\vec{AC}=2\vec{AB}$
• D. $\vec{BC}=3\vec{AB}$
• E. $\vec{AC}=4\vec{AB}$

Hint:

Collinear points satisfy $\vec{AC} = k\vec{AB}$.

Answer 1

Answer: (E)

To solve this problem, we need to evaluate the relationship between the position vectors of points \(A\), \(B\), and \(C\): \(\vec{OA}\), \(\vec{OB}\), and \(\vec{OC}\). We need to find the expressions for \(\vec{AB}\) and \(\vec{AC}\) and then examine their relationship.

1. Find \(\vec{AB}\): 
   \[ \vec{AB} = \vec{OB} - \vec{OA} = (3\hat{i} + \hat{j} - 2\hat{k}) - (2\hat{i} + 3\hat{j} - 5\hat{k}) \] Simplifying, we get: \(\vec{AB} = (3-2)\hat{i} + (1-3)\hat{j} + (-2+5)\hat{k} = \hat{i} - 2\hat{j} + 3\hat{k}.\)
2. Find \(\vec{AC}\): 
   \[ \vec{AC} = \vec{OC} - \vec{OA} = (6\hat{i} - 5\hat{j} + 7\hat{k}) - (2\hat{i} + 3\hat{j} - 5\hat{k}) \] Simplifying, we get: \(\vec{AC} = (6-2)\hat{i} + (-5-3)\hat{j} + (7+5)\hat{k} = 4\hat{i} - 8\hat{j} + 12\hat{k}.\)
3. Check the relationship \(\vec{AC} = n\vec{AB}\): 
   We need to determine the scalar \(n\) such that: 
   \[ \vec{AC} = n \cdot \vec{AB} = n(\hat{i} - 2\hat{j} + 3\hat{k}) \] Given \(\vec{AC} = 4\hat{i} - 8\hat{j} + 12\hat{k}\), compare components:
   • For the \(\hat{i}\) component: \(4 = n \cdot 1 \Rightarrow n = 4\)
   • For the \(\hat{j}\) component: \(-8 = n \cdot (-2) \Rightarrow n = 4\)
   • For the \(\hat{k}\) component: \(12 = n \cdot 3 \Rightarrow n = 4\)

After verifying the computations, the correct option is \(\vec{AC}=4\vec{AB}\).