If a, b and c are unit vectors, then |a - b |^2 + |b - c |^2 + |c… - Vidyadip

Question

If $a, b $ and $c $ are unit vectors, then $|a - b{|^2} + |b - c{|^2} + |c - a{|^2}$ does not exceed

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Answer

b
(b) $|a - b{|^2} + |b - c{|^2} + |c - a{|^2}$

$ = 2({a^2} + {b^2} + {c^2}) - 2(a.b + b\,.c + c\,.\,a)$

$ = 2 \times 3 - 2(a\,.\,b + b\,.\,c + c\,.\,a)$

$ = 6 - \{ {(a + b + c)^2} - {a^2} - {b^2} - {c^2}\} $$ = 9 - |a + b + c{|^2} \le 9.$

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