JEEOlympiad
The vectors a = (1, 2, 2), b = (2, -2, 1) and c = (2, 1, -2) are…
Question
The vectors a = (1, 2, 2), b = (2, -2, 1) and c = (2, 1, -2) are mutually perpendicular, each of length 3. A vector r of length 9 makes equal acute angles with a, b and c, and all of r's components in the a, b, c directions are positive. Find the value of (r·a)^2.
✓ Verified answer: 243checked by our engine — not a guess
Step-by-step solution
Since a, b, c are mutually orthogonal with equal length 3, the unit vectors ea = a/3, eb = b/3, ec = c/3 form an orthonormal basis.
Write r = p·ea + q·eb + s·ec.
Equal acute angles with all three (equal positive direction cosines) force p = q = s > 0. Then |r|^2 = p^2 + q^2 + s^2 = 3p^2 = 81, so p = q = s = 3√3.
Thus r = 3√3 (ea + eb + ec) and r·a = r·(3 ea) = 3·p = 9√3.
(r·a)^2 = (9√3)^2 = 81·3 = 243.
Final answer243
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