# 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

$\mathbf{a}^\perp = (a_x, a_y)^\perp = \text{Rot}(90^\circ)(a_x, a_y) = (-a_y, a_x)$

## 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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