For any two vectors and , prove that and hence show that (a1b1 + a2b2 + a3b3)2 ≤ (a12 + a22 + a32)(b12 + b22 + b32).

Question

For any two vectors  and , prove that  and hence show that (a1b1 + a2b2 + a3b3)2 ≤ (a12 + a22 + a32)(b12 + b22 + b32).

Philoid · Accepted Answer

To prove: and (a1b1 + a2b2 + a3b3)2 ≤ (a12 + a22 + a32)(b12 + b22 + b32)We know,From (1):Hence ProvedFormula used:Now,From (2):Hence Proved

For any two vectors and , prove that and hence show that (a₁b₁ + a₂b₂ + a₃b₃)² ≤ (a₁² + a₂² + a₃²)(b₁² + b₂² + b₃²).