Let a - b - c be three non coplanar vectors- Define v 1- v 2- v 3 bywhich of the following statements is false-A- a - v 1- b - v 2- c - v 3-0B- v 1-v-2-v-3-v-2-v-3-v-1-v-3-v-1-v-2-3-a-b-c-C- -v-1-v-2v-2-v-3v-3-v-1-1-b-c-2-D- -v1v2v3-1-a b-c-

The correct option is A a⃗.v⃗_⃗1⃗+b⃗.v⃗_⃗2⃗+c⃗.v⃗_⃗3⃗=0 Using [V̅_̅1̅V̅_̅2̅V̅_̅3̅]=[a̅×b̅b̅×c̅c̅×a̅]/[a̅b̅c̅]= [a̅b̅c̅]^2/[a̅b̅c̅]^3=1/[a̅b̅c̅] a̅.v̅_̅1̅+b̅.v ...

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Byju's Answer

Standard XII

Mathematics

Union of Sets

Let a , b , c...

Question

Let $¯ a, ¯ b, ¯ c$ be three non coplanar vectors. Define $_{1},_{2},_{3}$ by
$_{1} = \frac{\to b \times \to c}{[\to a \to b \to c]},_{2} = \frac{\to c \times \to a}{[\to a \to b \to c]},_{3} = \frac{\to a \times \to b}{[\to a \to b \to c]}$
which of the following statements is false?

\to a ._{1} + \to b ._{2} + \to c ._{3} = 0

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_{1} . (_{2} \times_{3}) +_{2} . (_{3} \times_{1}) +_{3} . (_{1} \times_{2}) = \frac{3}{[\to a \to b \to c]}

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[_{1}_{2}_{3}] = \frac{1}{[\to a \to b \to c]}

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[_{1} \times_{2}_{2} \times_{3}_{3} \times_{1}] = \frac{1}{[\to a \to b \to c]^{2}}

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Solution

The correct option is A $\to a ._{1} + \to b ._{2} + \to c ._{3} = 0$

Using
$[_{1}_{2}_{3}] = \frac{[¯ a \times ¯ b ¯ b \times ¯ c ¯ c \times ¯ a]}{[¯ a ¯ b ¯ c]} = \frac{[¯ a ¯ b ¯ c]^{2}}{[¯ a ¯ b ¯ c]^{3}} = \frac{1}{[¯ a ¯ b ¯ c]} ¯ a ._{1} + ¯ b ._{2} + ¯ c,_{3} = \sum \frac{¯ a . ¯ b \times ¯ c}{[¯ a ¯ b ¯ c]} = 3, [v_{1} v_{2} v_{3}] = \frac{1}{[¯ a ¯ b ¯ c]} \sum [v_{1} v_{2} v_{3}] = \frac{[a b c]^{2}}{[a b c]^{3}}$

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Let ¯a,¯b,¯c be three non coplanar vectors. Define →v1,→v2,→v3 by →v1=→b×→c[→a→b→c],→v2=→c×→a[→a→b→c],→v3=→a×→b[→a→b→c] which of the following statements is false?

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