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Multiplication with Vectors

If aα⃗×β⃗+b...

Question

If $a (\to α \times \to β) + b (\to β \times \to γ) + c (\to γ \times \to α) = \to 0$ at least one $a, b, c$ is non zero, then vectors, $\to α, \to β, \to γ$ are

A

Coplanar

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B

Mutually perpendicular

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C

Parallel

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D

None

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Solution

The correct option is A Coplanar
$a (\to α \times \to β) + b (\to α \times \to β) + c (\to γ \times \to α) = 0 a, b, c \to a t l e a s t o n e n o n - z e r o \Rightarrow a, b, c a l l c a n b e n o n - z e r o \Rightarrow E a c h t e r m s h o u l d b e e q u a l t o z e r o \Rightarrow \to α \times \to β = 0, \to α \times \to β = 0, \to γ \times \to α = 0 \Rightarrow α, β s h o u l d b e c o - p l a n a r β, γ s h o u l d b e c o - p l a n a r \Rightarrow α, β, γ a r e c o - p l a n a r O P T I O N : A$

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Similar questions

a (\to α \times \to β) + b (\to β \times \to γ) + c (\to γ \times \to α) = \to 0

and atleast one of

a, b, c

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\to α | | (\to β \times \to γ)

(\to α \times \to β) . (\to α \times \to γ)

\to α + \to β + \to γ = a \to δ

\to β + \to γ + \to δ = b \to α, \to α

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are non-collinear, then

\to α + \to β + \to γ + \to δ

\to α, \to β, \to γ

be three unit vectors such that

\to α . \to β = \to α . \to γ = 0

and the angle between

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