If vectors satisfy the condition |a - c| = |b - c|, then (b - a) … - Vidyadip

MCQ

If vectors satisfy the condition $|a - c| = |b - c|$, then $(b - a)\,.\,\left( {c - \frac{{a + b}}{2}} \right)$ is equal to

✓
$0$
B
$-1$
C
$1$
D
$2$

✓

Answer

Correct option: A.

$0$

a
(a) $(b - a)\,.\,\left( {c - \frac{{a + b}}{2}} \right) = b\,.\,c - b\,.\,\left( {\frac{{a + b}}{2}} \right)\, - a\,.\,c + \frac{a}{2}(a + b)$

and $|a - c|\, = \,|b - c|$ $ \Rightarrow $ $\,|a - c{|^2}\, = \,|b - c{|^2}$

$\therefore a + b = 2c$

Therefore, $(b - a).\,\left( {c - \frac{{a + b}}{2}} \right) = 0.$

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