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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 2**Question:** A marine robotics engineer is programming an autonomous underwater vehicle to optimize its path through a set of waypoints. The vehicle must move in such a way that its speed vector \(\mathbf{v}\) is always perpendicular to the vector \(\mathbf{w} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\). Find \(\mathbf{v}\) if it has a magnitude of 5 units.

📌 Solution:** To find a vector \(\mathbf{v}\) that is perpendicular to \(\mathbf{w} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\), we need \(\mathbf{v} \cdot \mathbf{w} = 0\). Let \(\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}\). The dot product condition gives:

📌 3x + 4y = 0

📌 We also know that the magnitude of \(\mathbf{v}\) is 5 units:

📌 \sqrt{x^2 + y^2} = 5 \quad \Rightarrow \quad x^2 + y^2 = 25

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