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Cauchy Schwartz Inequality stems from the simple looking beautiful identity.

\( (a^2 + b^2)(c^2 + d^2) = (ac – bd)^2 + (ad + bc)^2 \).

Now, using the trivial inequality that \( x^2 \geq 0 \), we get \( (a^2 + b^2)(c^2 + d^2) \geq (ad + bc)^2 \), when the equality holds if ad = bc rather \( \frac{a}{c} = \frac{b}{d}\).

This is the basis of Cauchy Schwartz Inequality. Similarly, it can be generalized to higher dimensions.

\( (a^2+b^2+c^2)(d^2+e^2+f^2)≥(ad+be+cf)^2 \). and equality holds iff \( \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \).