<p>A line segment has length 63 and direction ratios are 3, -2 and 6. Find the components of the line vector.</p>
Step-by-Step Solution
Key Concept: Direction ratios proportional to components; use length formula to find the scaling constant k.
Step 1: Let the components of the line segment on the axes be x, y, and z. Step 2: Using the given length: \(x^2 + y^2 + z^2 = 63^2\) Step 3: The direction ratios are proportional to components, so \(\frac{x}{3} = \frac{y}{-2} = \frac{z}{6} = k\) Step 4: Therefore, x = 3k, y = -2k, z = 6k Step 5: Substituting: \((3k)^2 + (-2k)^2 + (6k)^2 = 63^2\) Step 6: \(9k^2 + 4k^2 + 36k^2 = 3969\) Step 7: \(49k^2 = 3969\) Step 8: \(k = \pm 9\) Step 9: Components are (27, -18, 54) or (-27, 18, -54) ∴ Answer is (b) 27, -18, 54
Correct Answer: b