Question

fast answer ît is important

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Answer:

This follows from the Cauchy-Schwarz inequality.

$\left(\sum_{i=1}^n x_iy_i\right)^2 \leq \left(\sum_{i=1}^n x_i^2\right)\left(\sum_{i=1}^n y_i^2\right)\\\text{with equality if and only if there is a constant~ k such that y_i=kx_i for all~ i .}$

In terms of vectors

$|u\cdot v|^2 \leq |u|^2\,|v|^2 \ \text{with equality if and only if v=ku for some constant~ k .}$

Applying this to u = ( x₁, x₂, x₃ ) = ( a, b, c ) and v = ( y₁, y₂, y₃ ) = ( b, c, a ), we have

( ab + bc + ca )² ≤ ( a² + b² + c² ) ( b² + c² + a² ) = ( a² + b² + c² )²

with equality if and only if b = ka, c = kb, a = kc for some k.  But this would mean that a = kc = k²b = k³a, so k = 1 and therefore a = b = c.

The given condition says that a² + b² + c² = ab + bc + ca, so we do have equality above.  Therefore a = b = c.