If a, ,b, ,c are non-coplanar vectors and d = a + ,b + c, then is… - Vidyadip

MCQ

If $a,\,b,\,c$ are non-coplanar vectors and $d = \lambda a + \mu \,b + \nu c,$ then $\lambda $ is equal to

A
$\frac{{[d\,b\,c]}}{{[b\,a\,c]}}$
✓
$\frac{{[b\,c\,d]}}{{[b\,c\,a]}}$
C
$\frac{{[b\,d\,c]}}{{[a\,b\,c]}}$
D
$\frac{{[c\,b\,d\,]}}{{[a\,b\,c]}}$

✓

Answer

Correct option: B.

$\frac{{[b\,c\,d]}}{{[b\,c\,a]}}$

b
(b) Since $d=\lambda a+\mu b+\nu c$

$\therefore d.(b\times c)=\lambda \,a.(b\times c)+\mu \,b.(b\times c)+\mu \,c.(b\times c)$

$=\lambda \left[ abc \right]$

$⇒\lambda =\frac{[\,dbc]}{[abc]}$ $=\frac{[bcd]}{[bca]}$

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