A, B, C, D are any four points, thenAB , ,. , ,CD , , + , ,BC , ,… - Vidyadip

MCQ

$A, B, C, D $ are any four points, then$\overrightarrow {AB} \,\,.\,\,\overrightarrow {CD} \,\, + \,\overrightarrow {\,BC} \,\,.\,\,\overrightarrow {AD} \,\, + \overrightarrow {CA} \,\,.\,\,\overrightarrow {BD} \,\, = $

A
$2\,\,\overrightarrow {AB} \,\,.\,\,\overrightarrow {BC} \,\,.\,\,\overrightarrow {CD} $
B
$\overrightarrow {AB} \,\, + \,\,\overrightarrow {BC} \,\, + \,\,\overrightarrow {CD} $
C
$5\sqrt 3 $
✓
$0$

✓

Answer

Correct option: D.

$0$

d
(d) $\overrightarrow {AD} = \overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} = a + b + c$

$\overrightarrow {AC} = \overrightarrow {AB} + \overrightarrow {BC} = a + b$ or $\overrightarrow {CA} = - (a + b)$

$\overrightarrow {BD} = \overrightarrow {BC} + \overrightarrow {CD} = b + c$

Therefore, $\overrightarrow {AB} \,.\,\overrightarrow {CD} + \overrightarrow {BC} \,.\,\overrightarrow {AD} + \overrightarrow {CA} \,.\,\overrightarrow {BD} $

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

$ = a\,.\,c + b\,.\,a + b\,.\,b + b\,.\,c - a\,.\,b - a\,.\,c - b\,.\,b - b\,.\,c$

$ = 0$.

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