Show that the vectors ab ba and ab ba are orthogonalSoluti

Solution

For vectors to be orthogonal there dot product must be 0
Hence in this case we have
(||a||b + ||b||a).(||a||b - ||b||a) = |a|²b.b - |a|*a*|b|*b - |a|*a*|b|*b - |b|²a.a
Now a.a = |a|² and b.b = |b|²
Hence the equation converts to: |a|²|b|² - |a|a|b|b - |a|a|b|b - |b|²|a|² = 0
Hence the two vectors have dot product = 0
Hence they are orthogonal