<p>If three points A, B and C have position vectors \((1, x, 3)\), \((3, 4, 7)\) and \((y, -2, -5)\) respectively and if they are collinear, then \((x, y)\) is equal to</p>
Step-by-Step Solution
Key Concept: Three points are collinear if and only if the vectors formed by pairs of these points are parallel, which means one vector is a scalar multiple of the other. Equivalently, the cross product of vectors AB and AC equals zero.
Step 1: Form vectors from the given points Given: A = (1, x, 3), B = (3, 4, 7), C = (y, -2, -5) Vector AB = B - A = (3-1, 4-x, 7-3) = (2, 4-x, 4) Vector AC = C - A = (y-1, -2-x, -5-3) = (y-1, -2-x, -8) Step 2: Apply collinearity condition For collinear points, AC = λ·AB for some scalar λ Therefore: (y-1, -2-x, -8) = λ(2, 4-x, 4) Step 3: Set up component equations From the third component: -8 = 4λ ⟹ λ = -2 Step 4: Use λ = -2 in the first component y - 1 = λ(2) = -2(2) = -4 ⟹ y = -3 Step 5: Use λ = -2 in the second component -2 - x = λ(4 - x) = -2(4 - x) -2 - x = -8 + 2x -2 + 8 = 2x + x 6 = 3x ⟹ x = 2 Step 6: Verification With x = 2, y = -3: AB = (2, 2, 4) and AC = (-4, -4, -8) = -2(2, 2, 4) ✓ ∴ Answer: A
Correct Answer: A