If a-b-c are non-zero vectors such that a- b-c is perpendicular to a-b -c- then which of the following is-are true -

We have been given ((a× (b×c))·((a×b)×c)=0 ⇒ nbsp; nbsp;((a·c)b (a·b)c)·((a·c)b (b·c)a)=0 ⇒ (a·c)^2|b|^2 (a·c)(c·b)(a·b) (b·a)(a·c)(b·c)+(a·b)(b·c)(c·a)=0 ...

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Byju's Answer

Standard XII

Mathematics

Condition for Two Lines to be on the Same Plane

If a,b,c are ...

Question

If $\to a, \to b, \to c$ are non-zero vectors such that $\to a \times (\to b \times \to c)$ is perpendicular to $(\to a \times \to b) \times \to c,$ then which of the following is/are true $?$

\to a \cdot \to b = 0

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(\to a \cdot \to c) | \to b |^{2} = (\to a \cdot \to b) (\to b \cdot \to c)

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(\to a \cdot \to c) = 0

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\to b \cdot \to c = 0

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Solution

The correct option is C $(\to a \cdot \to c) = 0$
We have been given $((\to a \times (\to b \times \to c)) \cdot ((\to a \times \to b) \times \to c) = 0$ $\Rightarrow ((\to a \cdot \to c) \to b - (\to a \cdot \to b) \to c) \cdot ((\to a \cdot \to c) \to b - (\to b \cdot \to c) \to a) = 0$
$\Rightarrow (\to a \cdot \to c)^{2} | \to b |^{2} - (\to a \cdot \to c) (\to c \cdot \to b) (\to a \cdot \to b) -$ $(\to b \cdot \to a) (\to a \cdot \to c) (\to b \cdot \to c) + (\to a \cdot \to b) (\to b \cdot \to c) (\to c \cdot \to a) = 0$
$\Rightarrow (\to a \cdot \to c)^{2} | \to b |^{2} = (\to a \cdot \to c) (\to a \cdot \to b) (\to c \cdot \to b)$
$\Rightarrow (\to a \cdot \to c) = 0$ or
$(\to a \cdot \to c) | \to b |^{2} = (\to a \cdot \to b) (\to b \cdot \to c)$

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Condition for Two Lines to be on the Same Plane

Standard XII Mathematics

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