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Cross product: problem with breakdown of vector to parallel + orthogonal components

Vector $\vec{a}$ can be broken down into its components $\vec{a}_\parallel$ and $\vec{a}_\perp$ relative to $\vec{e}$. $\vec{a}_\parallel = (\vec{a}\vec{e})\vec{e}$ and $\vec{a}_\perp = ...