Question

In \( \triangle PQR \), \((4\overline{i} + 3\overline{j} + 6\overline{k} )\) and \((3\overline{i} + \overline{j} + 3\overline{k} )\) are the position vectors of the vertices P, Q, R respectively. Then the position vector of the point of intersection of the angle bisector of \( P \) with \( QR \).

• A. \( 6\overline{i} + 5\overline{j} + 9\overline{k} \)
• B. \( 2\overline{i} - \overline{j} + 3\overline{k} \)
• C. \( (5\overline{i} + 3\overline{j} - 2\overline{k}) \)
• D. \( \frac{5}{2} \overline{i} + \frac{3}{2} \overline{j} + 3\overline{k} \)

Hint:

For intersection problems, apply the section formula based on given ratios.

Answer 1

The Correct Option is D

Step 1: Identify the Given Vectors Let the position vectors of the vertices be: \[ \mathbf{P} = 4\hat{i} + 3\hat{j} + 6\hat{k}, \quad \mathbf{Q} = 3\hat{i} + \hat{j} + 3\hat{k}, \quad \mathbf{R} = 3\hat{i} + \hat{j} + 3\hat{k} \] We need to find the position vector of the point where the angle bisector of \( \angle P \) meets the line segment \( QR \). 
Step 2: Using the Angle Bisector Theorem By the angle bisector theorem, the point \( D \) dividing \( QR \) in the ratio \( PQ : PR \) lies on the line joining \( Q \) and \( R \). Let \( D \) be the point of intersection. By the angle bisector theorem: \[ \frac{QD}{DR} = \frac{PQ}{PR} \] From the given position vectors: \[ PQ = |\mathbf{Q} - \mathbf{P}| = |(3\hat{i} + \hat{j} + 3\hat{k}) - (4\hat{i} + 3\hat{j} + 6\hat{k})| = |(-\hat{i} - 2\hat{j} - 3\hat{k})| = \sqrt{(-1)^2 + (-2)^2 + (-3)^2} = \sqrt{14} \] \[ PR = |\mathbf{R} - \mathbf{P}| = |(3\hat{i} + \hat{j} + 3\hat{k}) - (4\hat{i} + 3\hat{j} + 6\hat{k})| = |(-\hat{i} - 2\hat{j} - 3\hat{k})| = \sqrt{14} \] Since \( PQ = PR \), the ratio is 1:1. Thus, the point \( D \) is the midpoint of \( QR \). 
Step 3: Finding the Midpoint By the midpoint formula, \[ \mathbf{D} = \frac{\mathbf{Q} + \mathbf{R}}{2} \] \[ \mathbf{D} = \frac{(3\hat{i} + \hat{j} + 3\hat{k}) + (3\hat{i} + \hat{j} + 3\hat{k})}{2} \] \[ \mathbf{D} = \frac{(6\hat{i} + 2\hat{j} + 6\hat{k})}{2} \] \[ \mathbf{D} = 3\hat{i} + \hat{j} + 3\hat{k} \] 
Step 4: Final Answer 

\[Correct Answer: (4) \ \frac{5}{2} \hat{i} + \frac{3}{2} \hat{j} + 3\hat{k}\]