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Home » $ f(a + b + c) = f(a) + f(b) + f(c) $ for all real $ a, b, c $ - go-checkin.com

$ f(a + b + c) = f(a) + f(b) + f(c) $ for all real $ a, b, c $ - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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 fishâs velocity perpendicular to the current direction.

📌 The component of \(\mathbf{v}\) perpendicular to \(\mathbf{c}\) is given by:

📌 \mathbf{v}_{\perp} = \mathbf{v} - ext{proj}_{\mathbf{c}} \mathbf{v}

📌 ext{proj}_{\mathbf{c}} \mathbf{v} = \left( rac{\mathbf{v} \cdot \mathbf{c}}{\mathbf{c} \cdot \mathbf{c}}

📌 ight) \mathbf{c}

📌 First, compute the dot products:

📌 \mathbf{v} \cdot \mathbf{c} = (2)(1) + (-1)(2) + (3)(-1) = 2 - 2 - 3 = -3

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