Slope of Orthogonal Curves

Theorem
Let $C_1$ and $C_2$ be curves in a cartesian plane.

Let $C_1$ and $C_2$ intersect each other at $P$.

Let the slope of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$.

Then $C_1$ and $C_2$ are orthogonal :
 * $m_1 = -\dfrac 1 {m_2}$

Proof
Let the slopes of $C_1$ and $C_2$ at $P$ be defined by the vectors $\mathbf v_1$ and $\mathbf v_2$ represented as column matrices:
 * $\mathbf v_1 = \begin{bmatrix} x_1 \\ y_1 \end{bmatrix}, \mathbf v_2 = \begin{bmatrix} x_2 \\ y_2 \end{bmatrix}$

By Non-Zero Vectors Orthogonal iff Perpendicular:
 * $\mathbf v_1 \cdot \mathbf v_2 = 0$ $C_1$ is orthogonal to $C_2$

where $\mathbf v_1 \cdot \mathbf v_2$ denotes the dot product of $C_1$ and $C_2$.

Thus: