If a-b-c be the three non-zero and non-collinar vectors such that a-b- c and b-c- a- then which of the following is-are true

Given a = b×c ⇒a= b× (a×b) (∵ nbsp;a×b=c) ⇒a= (b· nbsp; b)a (b· nbsp; a)b ⇒a = |b|^2a (b· nbsp; a)b Comparing both the sides, we get |b|^2 = 1 (b· ...

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Byju's Answer

Standard XII

Mathematics

Scalar Triple Product

If a,b,c be t...

Question

If $\to a, \to b, \to c$ be the three non-zero and non-collinar vectors such that $\to a \times \to b = \to c$ and $\to b \times \to c = \to a$ , then which of the following is/are true

\to a is perpendicular to \to b

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\to b is perpendicular to \to c

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| \to b | = 1

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| \to b | = 2

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Solution

The correct option is C $| \to b | = 1$
Given $\to a = \to b \times \to c$
$\Rightarrow \to a = \to b \times (\to a \times \to b)$
$(∵ \to a \times \to b = \to c)$
$\Rightarrow \to a = (\to b \cdot \to b) \to a - (\to b \cdot \to a) \to b \Rightarrow \to a = | \to b |^{2} \to a - (\to b \cdot \to a) \to b Comparing both the sides, we get | \to b |^{2} = 1 (\to b \cdot \to a) = 0 | \to b | = 1 and \to a \cdot \to b = 0 ∴ \to a is perpendicular to \to b$

$Now, \to c = \to a \times \to b = (\to b \times \to c) \times \to b = (\to b \cdot \to b) \to c - (\to c \cdot \to b) \to b \to c = | \to b |^{2} \to c - (\to c \cdot \to b) \to b On comparing both the sides | \to b |^{2} = 1 and (\to c \cdot \to b) = 0 ∴ \to b is perpendicular to \to c$

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If →a,→b,→c be the three non-zero and non-collinar vectors such that →a×→b=→c and →b×→c=→a, then which of the following is/are true

If $\to a, \to b, \to c$ be the three non-zero and non-collinar vectors such that $\to a \times \to b = \to c$ and $\to b \times \to c = \to a$ , then which of the following is/are true