If |a→|=1,|b→|=2,|c→|=3&a→,b→ are perpendicular to c→ and a→⋅b→=1,(a¯,b¯,c¯ form a vector traid in R.H.S) then a→×b→5b→×c→a→+b→+c→3=

a→×b→5b→×c→a→+b→+c→3=a→×b→53⋅b→×c→×a→+b→+c→=a→×b→53⋅[b→×c→×a→+b→×c→×b→+b→×c→×c→=a→×b→53⋅[ b.⇀a⇀c⇀−c→⋅a→b→+b→⋅b→c→−c→⋅b→ b→+b→⋅c→c→−c⇀⋅c→b→] =a→×b→53⋅[c→+4c→−9b→] (given a.→b →=1, a→.c→=0, b→.c→=0, also |a→|=1,|b→|=2,|c→|=3, a→2=a→.a→ )=a→×b→53.5c→-a→×b→53.9b → =a→×b→3.c→Now,a→⋅b→ =1 a→b→cosθ=1⇒1×2cos⁡θ=1⇒cos⁡θ=1/2⇒sin⁡θ=32a→× b→ =a→ b→n^sinθ⇒a→×b→=3n^⇒a→×b→3c→=3n^3c→=3

If |a→|=1,|b→|=2,|c→|=3&a→,b→ are perpendicular to c→ and a→⋅b→=1,(a¯,b¯,c¯ form a vector traid in R.H.S) then a→×b→5b→×c→a→+b→+c→3=

a→×b→5b→×c→a→+b→+c→3=a→×b→53⋅b→×c→×a→+b→+c→=a→×b→53⋅[b→×c→×a→+b→×c→×b→+b→×c→×c→=a→×b→53⋅[ b.⇀a⇀c⇀−c→⋅a→b→+b→⋅b→c→−c→⋅b→ b→+b→⋅c→c→−c⇀⋅c→b→] =a→×b→53⋅[c→+4c→−9b→] (given a.→b →=1, a→.c→=0, b→.c→=0, also |a→|=1,|b→|=2,|c→|=3, a→2=a→.a→ )=a→×b→53.5c→-a→×b→53.9b → =a→×b→3.c→Now,a→⋅b→ =1 a→b→cosθ=1⇒1×2cos⁡θ=1⇒cos⁡θ=1/2⇒sin⁡θ=32a→× b→ =a→ b→n^sinθ⇒a→×b→=3n^⇒a→×b→3c→=3n^3c→=3

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$If | \vec{a} | = 1, | \vec{b} | = 2, | \vec{c} | = 3 & \vec{a}, \vec{b} are perpendicular to \vec{c} and \vec{a} \cdot \vec{b} = 1, (\bar{a}, \bar{b}, \bar{c} form a vector traid$
$in R.H.S) then [\frac{\vec{a} \times \vec{b}}{5} \vec{b} \times \vec{c} \frac{\vec{a} + \vec{b} + \vec{c}}{\sqrt{3}}] =$

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$\sqrt{3}$

$3 \sqrt{3}$

answer is C.

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Detailed Solution

$\begin{array}{l} [\frac{\vec{a} \times \vec{b}}{5} \vec{b} \times \vec{c} \frac{\vec{a} + \vec{b} + \vec{c}}{\sqrt{3}}] \\ = \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} \cdot (\vec{b} \times \vec{c}) \times (\vec{a} + \vec{b} + \vec{c}) \\ = \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} \cdot [(\vec{b} \times \vec{c}) \times \vec{a} + (\vec{b} \times \vec{c}) \times \vec{b} + (\vec{b} \times \vec{c}) \times \vec{c}] \\ = \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} \cdot [(\overset{⇀}{b .} \overset{⇀}{a}) \overset{⇀}{c} - (\vec{c} \cdot \vec{a}) \vec{b} + (\vec{b} \cdot \vec{b}) \vec{c} - (\vec{c} \cdot \vec{b}) \vec{b} + (\vec{b} \cdot \vec{c}) \vec{c} - \overset{}{(\overset{⇀}{c} \cdot \vec{c}) \vec{b}]} \\ = \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} \cdot [\vec{c} + \vec{4 c} - 9 \vec{b}] (g i v e n \vec{a .} \vec{b} = 1, \vec{a} \vec{. c} = 0, \vec{b} . \vec{c} = 0, a l s o | \vec{a} | = 1, | \vec{b} | = 2, | \vec{c} | = 3, {|\vec{a}|}^{2} = \vec{a} . \vec{a}) \\ = \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} . 5 \vec{c} - \frac{\vec{a} \times \vec{b}}{5 \sqrt{3}} . \vec{9 b} = \frac{\vec{a} \times \vec{b}}{\sqrt{3}} \vec{. c} \\ Now, \overset{}{\vec{a} \cdot \vec{b}} = 1 |\vec{a}| |\vec{b}| \cos θ = 1 \\ \Rightarrow 1 \times 2 \cos θ = 1 \\ \Rightarrow \cos θ = 1 / 2 \Rightarrow \sin θ = \frac{\sqrt{3}}{2} \\ \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \hat{n} \sin θ \\ \Rightarrow \vec{a} \times \vec{b} = \sqrt{3} \hat{n} \Rightarrow \frac{\vec{a} \times \vec{b}}{\sqrt{3}} \vec{c} = \frac{\sqrt{3} \hat{n}}{\sqrt{3}} \vec{c} = 3 \end{array}$