Question

The co-ordinates of the points P and Q are (2, 6, 4) and (8, -3, 1) respectively. If the point R lies on the line segment PQ such that \(2|\overrightarrow{PR}|=|\overrightarrow{RQ}|\), then the co-ordinates of R are

• A. (4,-3,3)
• B. (4,3,-3)
• C. (2,-3,1)
• D. (4,3,3)
• E. (2,3,3)

Answer 1

The Correct Option is D

Step 1: Understand the section formula  
The given condition is: \[ 2 |\overrightarrow{PR}| = |\overrightarrow{RQ}| \] This means that the point \( R \) divides the line segment \( PQ \) in the ratio: \[ PR:RQ = 1:2 \] We use the section formula: \[ R \left( x, y, z \right) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right) \] where \( P(x_1, y_1, z_1) = (2, 6, 4) \) and \( Q(x_2, y_2, z_2) = (8, -3, 1) \), with the ratio \( m:n = 1:2 \).

Step 2: Compute the coordinates of \( R \) 
Using the section formula: 
\[ x = \frac{1(8) + 2(2)}{1+2} = \frac{8 + 4}{3} = \frac{12}{3} = 4 \] \[ y = \frac{1(-3) + 2(6)}{1+2} = \frac{-3 + 12}{3} = \frac{9}{3} = 3 \] \[ z = \frac{1(1) + 2(4)}{1+2} = \frac{1 + 8}{3} = \frac{9}{3} = 3 \]

Final Answer: The coordinates of \( R \) are (4,3,3).