Question

If the points with position vectors \[ x\vec i+2\vec j+y\vec k \] \[ \vec i-2\vec j+2x\vec k \] and \[ 2\vec i+3\vec j-\vec k \] are collinear, then \[ 10x-25y= \]

• A. \(-7\)
• B. \(20\)
• C. \(0\)
• D. \(1\)

Hint:

For collinear points in 3D, convert position vectors into coordinates and equate direction ratios.

Answer 1

The Correct Option is A

Concept: Three points are collinear when vectors formed are parallel. Coordinates: \[ P(x,2,y) \] \[ Q(1,-2,2x) \] \[ R(2,3,-1) \] Condition: \[ \frac{x_2-x_1}{x_3-x_2} = \frac{y_2-y_1}{y_3-y_2} = \frac{z_2-z_1}{z_3-z_2} \]

Step 1: Use first ratio.
\[ \frac{1-x}{2-1} = \frac{-2-2}{3+2} \] \[ 1-x=-\frac45 \] \[ x=\frac95 \]

Step 2: Use third ratio.
\[ \frac{2x-y}{-1-2x} = -\frac45 \] Substituting \[ x=\frac95 \] we get \[ y=1 \]

Step 3: Evaluate.
\[ 10x-25y \] \[ = 10\left(\frac95\right)-25 \] \[ = 18-25 \] \[ =-7 \] Thus \[ \boxed{-7} \]