2D Perp Operator

On the two dimensional plane, we can define the perp operator, that gives a counterclockwise (CCW) normal (i.e. perpendicular) of a vector \(\mathbf{a}\) by rotating the vector by 90°. The resulting vector is called the perp vector

\[\mathbf{a}^\perp = (a_x, a_y)^\perp = (-a_y, a_x)\]

Perp Operator Properties

Perpendicular

\[\mathbf{a}^\perp\cdot\mathbf{a}=0\]

Preserves length

\[|\mathbf{a}^\perp| = |\mathbf{a}|\]

Scalar Association

\[(\alpha\mathbf{a})^\perp = \alpha(\mathbf{a}^\perp) = \alpha\mathbf{a}^\perp\]

Linear

\[(\alpha\mathbf{a}+\beta\mathbf{b})^\perp = \alpha\mathbf{a}^\perp+\beta\mathbf{b}^\perp\]

Anti-potent

\[\mathbf{a}^{\perp\perp} = (\mathbf{a}^\perp)^\perp = -\mathbf{a}\]

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