Question

Let vectors: $ \vec{A} = \hat{i} - 2\hat{j} + \hat{k},\ \vec{B} = \hat{i} + \hat{j} - 2\hat{k},\ \vec{C} = 2\hat{i} - \hat{j},\ \vec{D} = \hat{i} + \hat{j} + \hat{k} $ If $ P $ divides $ AB $ in ratio 2:1 internally, and $ Q $ divides $ CD $ in ratio 1:2 externally, find the ratio in which the point $ 5\hat{i} - 6\hat{j} - 5\hat{k} $ divides line $ PQ $

• A. 2:1
• B. -2:1
• C. 2:3
• D. -2:3

Hint:

Use internal and external section formulas correctly to find points, then solve using vector equation.

Answer 1

The Correct Option is B

Step 1: Find coordinates of \( P \) dividing \( AB \) in 2:1 internally: \[ \vec{P} = \frac{2\vec{B} + 1\vec{A}}{3} \] Step 2: Find \( Q \) dividing \( CD \) in 1:2 externally: \[ \vec{Q} = \frac{1\vec{D} - 2\vec{C}}{1 - 2} \] Step 3: Now use section formula for vector: \[ \vec{R} = \frac{m\vec{Q} + n\vec{P}}{m + n} = 5\hat{i} - 6\hat{j} - 5\hat{k} \Rightarrow \text{Solve to find ratio } m:n = -2:1 \]