Question

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

• A. $\frac{\lbrack\mathbf{dbc}\rbrack}{\lbrack\mathbf{bac}\rbrack}$
• B. $\frac{\lbrack\mathbf{bcd}\rbrack}{\lbrack\mathbf{bca}\rbrack}$
• C. $\frac{\lbrack\mathbf{bdc}\rbrack}{\lbrack\mathbf{abc}\rbrack}$
• D. $\frac{\lbrack\mathbf{cbd}\rbrack}{\lbrack\mathbf{abc}\rbrack}$

Answer 1

Answer: (B)

$\frac{\lbrack\mathbf{bcd}\rbrack}{\lbrack\mathbf{bca}\rbrack}$

Answer 2

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

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

$= \lambda\lbrack abc\rbrack$

⇒ $\lambda = \frac{\lbrack dbc\rbrack}{\lbrack abc\rbrack} = \frac{\lbrack bcd\rbrack}{\lbrack bca\rbrack}$.