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