Let a→ , b→ and c→ be three unit vectors such that |a→+b→+c→|=3 and (a→×b→) . (b→×c→)+(b→×c→) . (c→×a→) +(c→×a→) . (a→×b→)=λ.Which of the following are CORRECT ?

If λ is maximum then the volume of parallelepiped determined by a ¯ × b ¯ , 2 b ¯ × c ¯ and c ¯ × a ¯ is 2. , If λ is maximum then the volume of parallelepiped determined by a → , b → and c → is 3 , The maximum value of λ is 0

Let a→ , b→ and c→ be three unit vectors such that |a→+b→+c→|=3 and (a→×b→) . (b→×c→)+(b→×c→) . (c→×a→) +(c→×a→) . (a→×b→)=λ.Which of the following are CORRECT ?

( a → × b → ) . ( b → × c → ) = ( b → . c → ) ( a → . b → ) − a → . c → ( b → × c → ) . ( c → × a → ) = ( a → . c → ) ( b → . c → ) − a → . b → ( c → × a → ) . ( a → × b → ) = ( a → . b → ) ( a → . c → ) − b → . c → Given that | a → + b → + c → | = 3 ⇒ ( a → + b → + c → ) . ( a → + b → + c → ) = 3 ⇒ a → . b → + b → . c → + c → . a → = 0 λ = ( a → . b → ) . ( b → . c → ) + ( b → . c → ) ( c → . a → ) + ( c → . a → ) ( a → . b → ) − a → . b → − b → . c → − c → . a → ⇒ λ = ( a → . b → ) . ( b → . c → ) + ( b → . c → ) ( c → . a → ) + ( c → . a → ) ( a → . b → ) ⇒ λ ≤ 0 [since, x + y + z = 0, xy + yz + zx ≤ 0] λ max = 0 only when a → . b → = b → . c → = a → . c → = 0 ⇒ a → ⊥ b , b → ⊥ c → and c → ⊥ a → ( 2 a → + 3 b → + 4 c → ) . ( a → × b → + 5 b → × c → + 6 c → × a → ) = 10 a → . ( b → × c → ) + 18 b → . ( c → × a → ) + 4 c → . ( a → × b → ) = 32

If λ is maximum then the value of | ( 2 a → + 3 b → + 4 c → ) . ( a → × b → + 5 b → × c → + 6 c → × a → ) | is 32

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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $| \vec{a} + \vec{b} + \vec{c} | = \sqrt{3}$ and $(\vec{a} \times \vec{b}) . (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) . (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) . (\vec{a} \times \vec{b}) = λ$ .Which of the following are CORRECT ?

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If $λ$ is maximum then the volume of parallelepiped determined by $\bar{a} \times \bar{b}, 2 \bar{b} \times \bar{c}$ and $\bar{c} \times \bar{a}$ is 2.

If $λ$ is maximum then the value of $| (2 \vec{a} + 3 \vec{b} + 4 \vec{c}) . (\vec{a} \times \vec{b} + 5 \vec{b} \times \vec{c} + 6 \vec{c} \times \vec{a}) |$ is 32

If $λ$ is maximum then the volume of parallelepiped determined by $\vec{a}, \vec{b}$ and $\vec{c}$ is $\sqrt{3}$

The maximum value of $λ$ is 0

answer is A, C, D.

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Detailed Solution

$(\vec{a} \times \vec{b}) . (\vec{b} \times \vec{c}) = (\vec{b} . \vec{c}) (\vec{a} . \vec{b}) - \vec{a} . \vec{c}$
$(\vec{b} \times \vec{c}) . (\vec{c} \times \vec{a}) = (\vec{a} . \vec{c}) (\vec{b} . \vec{c}) - \vec{a} . \vec{b}$
$(\vec{c} \times \vec{a}) . (\vec{a} \times \vec{b}) = (\vec{a} . \vec{b}) (\vec{a} . \vec{c}) - \vec{b} . \vec{c}$
Given that
$| \vec{a} + \vec{b} + \vec{c} | = \sqrt{3} \Rightarrow (\vec{a} + \vec{b} + \vec{c}) . (\vec{a} + \vec{b} + \vec{c}) = 3 \Rightarrow \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} = 0$
$λ = (\vec{a} . \vec{b}) . (\vec{b} . \vec{c}) + (\vec{b} . \vec{c}) (\vec{c} . \vec{a}) + (\vec{c} . \vec{a}) (\vec{a} . \vec{b}) - \vec{a} . \vec{b} - \vec{b} . \vec{c} - \vec{c} . \vec{a}$
$\Rightarrow λ = (\vec{a} . \vec{b}) . (\vec{b} . \vec{c}) + (\vec{b} . \vec{c}) (\vec{c} . \vec{a}) + (\vec{c} . \vec{a}) (\vec{a} . \vec{b})$
$\Rightarrow λ \leq 0$ [since, x + y + z = 0, xy + yz + zx $\leq$ 0]
$λ_{\max}$ = 0 only when $\vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{a} . \vec{c} = 0$
$\Rightarrow \vec{a} ⊥ b, \vec{b} ⊥ \vec{c}$ and $\vec{c} ⊥ \vec{a}$
$(2 \vec{a} + 3 \vec{b} + 4 \vec{c}) . (\vec{a} \times \vec{b} + 5 \vec{b} \times \vec{c} + 6 \vec{c} \times \vec{a})$
= $10 \vec{a} . (\vec{b} \times \vec{c}) + 18 \vec{b} . (\vec{c} \times \vec{a}) + 4 \vec{c} . (\vec{a} \times \vec{b}) = 32$

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