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Problem The volume of the parallelopiped whose coterminous edges are represented by the vectors 2 b × c 2\mathbf{b} \times \mathbf{c} 2 b × c 3 c × a 3\mathbf{c} \times \mathbf{a} 3 c × a 4 a × b 4\mathbf{a} \times \mathbf{b} 4 a × b a = ( 1 + sin θ ) i ^ + cos θ j ^ + sin 2 θ k ^ \mathbf{a} = (1 + \sin \theta)\hat{\mathbf{i}} + \cos \theta \hat{\mathbf{j}} + \sin 2\theta \hat{\mathbf{k}} a = ( 1 + sin θ ) i ^ + cos θ j ^ + sin 2 θ k ^ b = sin ( θ + 2 π 3 ) i ^ + cos ( θ + 2 π 3 ) j ^ + sin ( 2 θ + 4 π 3 ) k ^ , \mathbf{b} = \sin\left(\theta + \frac{2\pi}{3}\right)\hat{\mathbf{i}} + \cos\left(\theta + \frac{2\pi}{3}\right)\hat{\mathbf{j}} + \sin\left(2\theta + \frac{4\pi}{3}\right)\hat{\mathbf{k}}, b = sin ( θ + 3 2 π ) i ^ + cos ( θ + 3 2 π ) j ^ + sin ( 2 θ + 3 4 π ) k ^ , c = sin ( θ − 2 π 3 ) i ^ + cos ( θ − 2 π 3 ) j ^ + sin ( 2 θ − 4 π 3 ) k ^ is 18 \mathbf{c} = \sin\left(\theta - \frac{2\pi}{3}\right)\hat{\mathbf{i}} + \cos\left(\theta - \frac{2\pi}{3}\right)\hat{\mathbf{j}} + \sin\left(2\theta - \frac{4\pi}{3}\right)\hat{\mathbf{k}} \text{ is 18} c = sin ( θ − 3 2 π ) i ^ + cos ( θ − 3 2 π ) j ^ + sin ( 2 θ − 3 4 π ) k ^ is 18 θ \theta θ [ 0 , π 2 ] \left[0, \frac{\pi}{2}\right] [ 0 , 2 π ] π 9 \frac{\pi}{9} 9 π 2 π 9 \frac{2\pi}{9} 9 2 π π 3 \frac{\pi}{3} 3 π 4 π 9 \frac{4\pi}{9} 9 4 π Browse the problem library