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Scalar Triple Product

If a⃗, b⃗, c⃗...

Question

If $\to a, \to b, \to c$ are non-coplanar vectors and $\to d = λ \to a + μ \to b + ν \to c,$ then $λ$ equal to

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is B
$\frac{[\to b \to c \to d]}{[\to b \to c \to a]}$

$\to d = λ \to a + μ \to b + ν \to c ∴ [\to d \to b \to c] = \to d . \to b \times \to c = λ \to a . \to b \times \to c + μ \to b . \to b \times \to c + ν \to c . \to b \times \to c = λ [\to a \to b \to c] + 0 + 0 = λ [\to a \to b \to c] \Rightarrow λ = \frac{[\to d \to b \to c]}{[\to a \to b \to c]} = \frac{[\to b \to c \to d]}{[\to b \to c \to a]}$

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Similar questions

\to a, \to b, \to c, \to d

are non-coplanar vectors then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

is parallel to:

\to b, \to c, \to d

are non-coplanar vectors then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

\to b, \to c, \to d

are non-coplanar vectors, then the vector

(\to a \times \to b) \times (\to c \times \to d) + (\to a \times \to c) \times (\to d \times \to b) + (\to a \times \to d) \times (\to b \times \to c)

\to a, \to b, \to c

are unit vectors such that

\to a . \to b = 0

(\to a - \to c) . (\to b + \to c) = 0

\to c = λ \to a + μ \to b + ω (\to a \times \to b)

for some scalars

λ, μ, ω

\to a, \to b, \to c

are unit vectors such that

\to a \cdot \to b = 0

(\to a - \to c) \cdot (\to b + \to c) = 0

\to c = λ \to a + μ \to b + ω (\to a \times \to b)

λ, μ, ω

are scalars then?

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