Let a, b, c be unit vectors. Further, a is perpendicular to b; c makes an angle π/3 with both a and b; and c = pa + qb + r(a×b). What is the value of r²?

Since |c|² = 1 and c = pa + qb + r(a×b) with a, b, a×b mutually perpendicular and |a×b| = |a||b|sin(π/2) = 1. So |c|² = p² + q² + r²|a×b|² = p² + q² + r². With p = q = 1/2: 1 = 1/4 + 1/4 + r², giving r² = 1 - 1/2 = 1/2.