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Let r = sin(x) * (a x b) + cos(y) * (b x c) + 2 * (c x a), where a, b, c are three non-collinear vectors. It is given that r is perpendicular to (a + b + c). Which of the following are possible values of x² + y²?
- pi²
- 5 * pi² / 4
- 35 * pi² / 4
- 37 * pi² / 4
Correct answer: 5 * pi² / 4
Solution
The perpendicularity condition forces sin(x) = -1 and cos(y) = -1. The general solutions give x = -pi/2 + 2k*pi and y = (2m+1)*pi. Computing x² + y² for the smallest values and other combinations yields possible values including 5*pi²/4 and 37*pi²/4.
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