An ichthyologist is modeling the movement of a fish in a 3D aquatic environment using vector dynamics. Suppose the fish's velocity vector is \(\mathbf{v} = egin{pmatrix} 2 \ -1 \ 3 \end{pmatrix}\) and the direction of the current is given by \(\mathbf{c} = egin{pmatrix} 1 \ 2 \ -1 \end{pmatrix}\). Find the component of the fish’s velocity perpendicular to the current direction.

["Title: Finding the Perpendicular Component of a Fish’s Velocity to Current Direction Using Vector Dynamics", "In ichthyology, modeling fish movement in 3D aquatic environments often requires breaking vectors into meaningful components, especially when analyzing how swimming motion interacts with environmental currents. A common problem involves determining the part of a fish’s velocity vector that is perpendicular to the direction of a water current—this perpendicular component affects how the fish maneuvers and maintains position despite flowing waters.", "Given the fish’s velocity vector\n[\n\mathbf{v} = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix}\n]\nand the current direction vector\n[\n\mathbf{c} = \begin{pmatrix} 1 \ 2 \ -1 \end{pmatrix},\n]\nwe aim to compute the component of (\mathbf{v}) perpendicular to (\mathbf{c}). This perpendicular component captures the fish’s true swimming motion independent of the flow, crucial for understanding locomotion and energy expenditure.", "### Step 1: The projection of (\mathbf{v}) onto (\mathbf{c})", "The projection of (\mathbf{v}) in the direction of (\mathbf{c}) is found using the dot product:\n[\n\mathbf{v}{\parallel} = \frac{\mathbf{v} \cdot \mathbf{c}}{|\mathbf{c}|^2} \mathbf{c}\n]\nFirst, compute the dot product:\n[\n\mathbf{v} \cdot \mathbf{c} = (2)(1) + (-1)(2) + (3)(-1) = 2 - 2 - 3 = -3\n]\nNext, the magnitude squared of (\mathbf{c}):\n[\n|\mathbf{c}|^2 = 1^2 + 2^2 + (-1)^2 = 1 + 4 + 1 = 6\n]\nThus, the projection vector is:\n[\n\mathbf{v}} = \frac{-3}{6} \begin{pmatrix} 1 \ 2 \ -1 \end{pmatrix} = \begin{pmatrix} -0.5 \ -1 \ 0.5 \end{pmatrix\n]", "### Step 2: The perpendicular component", "The perpendicular component (\mathbf{v}{\perp}) is obtained by subtracting the parallel projection from the total velocity:\n[\n\mathbf{v}} = \mathbf{v} - \mathbf{v{\parallel} = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix} - \begin{pmatrix} -0.5 \ -1 \ 0.5 \end{pmatrix} = \begin{pmatrix} 2.5 \ 0 \ 2.5 \end{pmatrix}\n]", "Alternatively, using vector identity:\n[\n\mathbf{v}} = \mathbf{v} - \frac{\mathbf{v} \cdot \mathbf{c}}{|\mathbf{c}|^2} \mathbf{c\n]\nwhich confirms our result.", "### Conclusion", "The component of the fish’s velocity perpendicular to the current is:\n[\n\boxed{\begin{pmatrix} 2.5 \ 0 \ 2.5 \end{pmatrix}}\n]", "Modeling this perpendicular velocity component helps ichthyologists analyze how fish actively counteract currents to maintain direction and stability—key insights for conservation, taxonomy, and underwater robotics inspired by fish behavior.", "Keywords: Ichthyology, fish velocity, vector dynamics, current direction, perpendicular component, projective vector analysis, aquatic locomotion"]