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

MCQ

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

A
$\frac{[ dbc ]}{[ bac ]}$
✓
$\frac{[b c d]}{[b c a]}$
C
$\frac{[ bdc ]}{[ abc ]}$
D
$\frac{[ cbd ]}{[ abc ]}$

✓

Answer

Correct option: B.

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

(B) Since $d=\lambda a+\mu b+v c$
$\therefore \quad d \cdot(b \times c)=\lambda a \cdot(b \times c)+\mu b \cdot(b \times c)+v c \cdot(b \times c)$
$\Rightarrow d \cdot( b \times c )=\lambda[ a b c ]$
$\Rightarrow \lambda=\frac{[ dbc ]}{[ abc ]}=\frac{[ bcd ]}{[ bca ]}$

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