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Question: An anthropologist studying ritualistic circular dance formations interprets rhythmic movements as vectors; find the vector $ \mathbf{v} = \langle x, y, z \rangle $ satisfying $ \mathbf{v} \times \langle 1, 2, 3 \rangle = \langle 1, -2, 1 \rangle $ and $ \mathbf{v} \cdot \langle 1, 2, 3 \rangle = 0 $.
Solution: Let $ \mathbf{a} = \langle 1, 2, 3 \rangle $, $ \mathbf{b} = \langle 1, -2, 1 \rangle $. We are to find $ \mathbf{v} $ such that:
\mathbf{v} \times \mathbf{a} = \mathbf{b}, \quad \mathbf{v} \cdot \mathbf{a} = 0

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