The volume of the parallelopiped whose coterminous edges are represented by the vectors 2b→×c→,3c→×a→ and 4a→×b→ where a→=1+sinθi^+cosθj^+sin2θk^, b→=sinθ+2π3i^+cosθ+2π3j^+sin2θ+4π3k^, c→=sinθ−2π3i^+cosθ−2π3j^+sin2θ−4π3k^ is 18 cubic units, then the sum values of θ in the interval 0,π2, is

The volume of the parallelopiped whose coterminous edges are represented by the vectors 2b→×c→,3c→×a→ and 4a→×b→ where a→=1+sinθi^+cosθj^+sin2θk^, b→=sinθ+2π3i^+cosθ+2π3j^+sin2θ+4π3k^, c→=sinθ−2π3i^+cosθ−2π3j^+sin2θ−4π3k^ is 18 cubic units, then the sum values of θ in the interval 0,π2, is

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The volume of the parallelopiped whose coterminous edges are represented by the vectors $2 \vec{b} \times \vec{c}, 3 \vec{c} \times \vec{a}$ and $4 \vec{a} \times \vec{b}$ where $\vec{a} = (1 + \sin θ) \hat{i} + \cos θ \hat{j} + \sin 2 θ \hat{k},$ $\vec{b} = \sin (θ + \frac{2 π}{3}) \hat{i} + \cos (θ + \frac{2 π}{3}) \hat{j} + \sin (2 θ + \frac{4 π}{3}) \hat{k}$ , $\vec{c} = \sin (θ - \frac{2 π}{3}) \hat{i} + \cos (θ - \frac{2 π}{3}) \hat{j} + \sin (2 θ - \frac{4 π}{3}) \hat{k}$ is 18 cubic units, then the sum values of $θ$ in the interval $(0, \frac{π}{2})$ , is

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$\frac{π}{9}$

$\frac{2 π}{9}$

$\frac{4 π}{9}$

$\frac{7 π}{9}$

answer is C.

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Detailed Solution

$Volume = |[\begin{array}{lll} 2 b \times c 3 c \times a 4 a \times b \end{array}]| = 18$

$24 [a b c]^{2} = 18$

$| [a b c] | = \frac{\sqrt{3}}{2}$

$Now, [a b c ] = |\begin{array}{ccc} (1 + \sin θ) & \cos θ & \sin 2 θ \\ \sin (θ + \frac{2 π}{3}) & \cos (θ + \frac{2 π}{3}) & \sin (2 θ + \frac{4 π}{3}) \\ \sin (θ - \frac{2 π}{3}) & \cos (θ - \frac{2 π}{3}) & \sin (2 θ - \frac{4 π}{3}) \end{array}|$