On the two-dimensional plane, we can define the perp operator, which gives a counterclockwise (CCW) normal vector (i.e. perpendicular) of a vector \(\mathbf{a}\) by rotating it by \(90^\circ\). The resulting vector is called the perp vector.
\[ \mathbf{a}^\perp = (a_x, a_y)^\perp = \operatorname{Rot}(90^\circ)(a_x, a_y) = (-a_y, a_x) \]
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:
\[\mathbf{a}^{\perp\perp} = (\mathbf{a}^\perp)^\perp = -\mathbf{a}\]