If a-b and c are non-coplanar vectors and a-c is perpendicular to a- b-c- then the value of -a- b-c-c is equal to

Given nbsp;(a×c) is perpendicular to (a× (b×c)) ⇒(a× (b×c))·(a×c)=0 ⇒ [(a·c)b (a·b)c]· (a×c)=0 ⇒ (a·c)[b a c]=0 (As nbsp;a,b and c are non coplanar vectors. ...

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Standard XII

Mathematics

Scalar Triple Product

If a,b and c ...

Question

$If \to a, \to b and \to c$ are non-coplanar vectors and $\to a \times \to c$ is perpendicular to $\to a \times (\to b \times \to c),$ then the value of $[\to a \times (\to b \times \to c)] \times \to c$ is equal to

[\to a \to b \to c] \to c

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[\to a \to b \to c] \to b

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\to 0

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[\to a \to b \to c] \to a

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Solution

The correct option is C $\to 0$
Given $(\to a \times \to c)$ is perpendicular to $(\to a \times (\to b \times \to c)) \Rightarrow (\to a \times (\to b \times \to c)) \cdot (\to a \times \to c) = 0$ $\Rightarrow [(\to a \cdot \to c) \to b - (\to a \cdot \to b) \to c] \cdot (\to a \times \to c) = 0 \Rightarrow (\to a \cdot \to c) [\to b \to a \to c] = 0$ (As $\to a, \to b$ and $\to c$ are non-coplanar vectors.
So, $[\to b \to a \to c] \neq 0$ )
$∴ (\to a \cdot \to c) = 0$

Now, $\to a \times (\to b \times \to c) = (\to a \cdot \to c) \to b - (\to a \cdot \to b) \to c$
$\Rightarrow \to a \times (\to b \times \to c) = - (\to a \cdot \to b) \to c$
$\Rightarrow [\to a \times (\to b \times \to c)] \times \to c = - (\to a \cdot \to b) (\to c \times \to c) = \to 0$

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Scalar Triple Product

Standard XII Mathematics

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If →a,→b and →c are non-coplanar vectors and →a×→c is perpendicular to →a×(→b×→c), then the value of [→a×(→b×→c)]×→c is equal to

$If \to a, \to b and \to c$ are non-coplanar vectors and $\to a \times \to c$ is perpendicular to $\to a \times (\to b \times \to c),$ then the value of $[\to a \times (\to b \times \to c)] \times \to c$ is equal to