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Condition for Two Lines to be on the Same Plane

If αa + βb + ...

Question

If $α \to a + β \to b + γ \to c = \to 0,$ then the resultant of $(\to a \times \to b) \times {(\to b \times \to c) \times (\to c \times \to a)}$ is
(where $α, β$ and $γ$ are not simultaneously zero.)

A

α - β + γ

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B

(α + β + γ) (\to a + \to b + \to c)

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C

γ \to a + \to b

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D

\to 0

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Solution

The correct option is D $\to 0$
As $α \to a + β \to b + γ \to c = \to 0$
Therefore, the vectors $\to a, \to b, \to c$ are coplanar.
$\Rightarrow \to b \times \to c$ and $\to c \times \to a$ are parallel
$\Rightarrow (\to b \times \to c) \times (\to c \times \to a) = \to 0$
Hence, $(\to a \times \to b) \times {(\to b \times \to c) \times (\to c \times \to a)}$
$= (\to a \times \to b) \times {\to 0} = \to 0$

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Condition for Two Lines to be on the Same Plane

Standard XII Mathematics

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