Let a = a₁ i + a₂ j + a₃ k and b = b₁ i + b₂ j + b₃ k be two non-zero vectors with real components. If M is the maximum possible value of the expression a b | a | (|b₁|, |b₂|, |b₃|) , then the value of M^2 is

Step-by-step solution

Let the given expression be E = a b | a | (|b₁|, |b₂|, |b₃|) . By the Cauchy-Schwarz inequality (or the definition of the dot product), we know that a b | a | | b | . The equality holds when a and b are parallel and in the same direction. Using this upper bound, we can write: E | a | | b | | a | (|b₁|, |b₂|, |b₃|) = | b | (|b₁|, |b₂|, |b₃|) Now, let k = (|b₁|, |b₂|, |b₃|) . Then |b₁| k , |b₂| k , and |b₃| k . The magnitude of b is given by | b | = b₁^2 + b₂^2 + b₃^2 . Since each squared component is at most k^2 , w