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.

["Optimizing Autonomous Vehicle Path: Determining a Speed Vector Perpendicular to a Given Direction with Fixed Magnitude", "When programming an autonomous underwater vehicle (AUV) to follow a precise trajectory through underwater waypoints, engineers must balance speed, direction, and energy efficiency. A critical challenge arises when the vehicle’s velocity vector (\mathbf{v}) must remain perpendicular to a fixed reference vector (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}), while maintaining a constant magnitude of 5 units. Understanding how to construct such a vector is essential for optimal path planning and motion control.", "## The Perpendicular Velocity Vector Constraint", "In vector mathematics, a vector (\mathbf{v} = \begin{bmatrix} v_x \ v_y \end{bmatrix}) is perpendicular to (\mathbf{w} = \begin{bmatrix} 3 \ 4 \end{bmatrix}) if their dot product equals zero:", "[\n\mathbf{v} \cdot \mathbf{w} = 3v_x + 4v_y = 0\n]", "This condition ensures smooth, non-clash path segments in the vehicle’s travel—especially important in complex underwater environments with obstacles and tight maneuvering constraints.", "## Finding a Candidate Vector with Desired Magnitude", "We seek (\mathbf{v}) such that:\n1. (\mathbf{v} \cdot \mathbf{w} = 0) (perpendicularity),\n2. (|\mathbf{v}| = 5) (magnitude constraint).", "Let’s choose one free variable to satisfy both conditions.", "From the perpendicularity equation:", "[\n3v_x + 4v_y = 0 \quad \Rightarrow \quad v_y = -\frac{3}{4}v_x\n]", "Substitute into the magnitude condition:", "[\n|\mathbf{v}|^2 = v_x^2 + v_y^2 = 25\n]", "[\nv_x^2 + \left(-\frac{3}{4}v_x\right)^2 = 25\n]\n[\nv_x^2 + \frac{9}{16}v_x^2 = 25\n]\n[\n\left(1 + \frac{9}{16}\right)v_x^2 = 25\n]\n[\n\frac{25}{16}v_x^2 = 25\n]\n[\nv_x^2 = 16 \quad \Rightarrow \quad v_x = \pm 4\n]", "Take (v_x = 4) → then (v_y = -\frac{3}{4}(4) = -3)", "Thus:", "[\n\mathbf{v}_1 = \begin{bmatrix} 4 \ -3 \end{bmatrix}\n]", "Check magnitude:", "[\n|\mathbf{v}_1| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \quad \ ext{✓}\n]", "Check perpendicularity:", "[\n3(4) + 4(-3) = 12 - 12 = 0 \quad \ ext{✓}\n]", "Alternatively, the opposite direction also satisfies perpendicularity:", "[\n\mathbf{v}_2 = \begin{bmatrix} -4 \ 3 \end{bmatrix}\n]", "Both vectors are valid solutions satisfying the constraint of perpendicular motion with magnitude 5.", "## Application in Marine Robotics Path Planning", "For an autonomous underwater vehicle navigating between waypoints, ensuring the velocity vector is perpendicular to a reference direction (e.g., current vector (\mathbf{w})) can minimize sideways drift, reduce energy loss from cross-drag forces, and improve trajectory accuracy—critical in deep-sea exploration or inspection missions.", "Programmers can precompute such vectors to generate smooth, efficient paths aligned with environmental flow or mission constraints. Leveraging perpendicularity ensures motion components remain optimally aligned for propulsion efficiency, especially in constrained or obstacle-dense zones.", "---", "Conclusion:\nTo keep an underwater vehicle’s velocity vector (\mathbf{v}) perpendicular to (\begin{bmatrix} 3 \ 4 \end{bmatrix}) while maintaining a speed of 5 units, two valid solutions are:", "[\n\mathbf{v} = \begin{bmatrix} 4 \ -3 \end{bmatrix} \quad \ ext{or} \quad \mathbf{v} = \begin{bmatrix} -4 \ 3 \end{bmatrix}\n]", "These vectors enable precise, energy-efficient navigation—vital for modern marine robotics. Understanding vector geometry empowers engineers to optimize vehicle dynamics in complex underwater environments."]