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our deepest fear is not that we are inadequate - inBeat

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 \cos \phi = \sqrt{1 - rac{\sin^2 heta}{n^2}}

📌 Thus, the expression for \(\cos \phi\) is:

📌 oxed{\sqrt{1 - rac{\sin^2 heta}{n^2}}}

📌 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}}

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