Question

Let point $ P $ divide the line segment joining $ A(1, 2, -1) $ and $ B(-1, 0, 1) $ externally in the ratio $ 1:2 $. Given point $ Q = (1, 3, -1) $, find $ PQ $.

• A. \( \sqrt{10} \)
• B. \( 3 \)
• C. \( 1 \)
• D. \( \sqrt{13} \)

Hint:

For external division in 3D, apply section formula carefully with negative sign in denominator.

Answer 1

The Correct Option is B

Use external division formula: \[ P = \left( \frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n}, \frac{m z_2 - n z_1}{m - n} \right) \] Let \( A = (1, 2, -1),\ B = (-1, 0, 1),\ m:n = 1:2 \) Then, \[ P = \left( \frac{1 \cdot (-1) - 2 \cdot 1}{1 - 2}, \frac{1 \cdot 0 - 2 \cdot 2}{1 - 2}, \frac{1 \cdot 1 - 2 \cdot (-1)}{1 - 2} \right) = \left( \frac{-1 - 2}{-1}, \frac{-4}{-1}, \frac{1 + 2}{-1} \right) = (3, 4, -3) \] Point \( Q = (1, 3, -1) \) Now, \[ PQ = \sqrt{(3 - 1)^2 + (4 - 3)^2 + (-3 + 1)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = \boxed{3} \]