Parametric Equations and Vectors

Parametric Equations

Instead of $y = f(x)$, both $x$ and $y$ depend on a parameter $t$:

$x = f(t), \quad y = g(t)$

Eliminating the Parameter

1. Solve one equation for $t$
2. Substitute into the other

Example: $x = 2t + 1$, $y = t^2$ $t = \frac{x-1}{2} \implies y = \left(\frac{x-1}{2}\right)^2 = \frac{(x-1)^2}{4}$

Parametric Line

$x = x_0 + at, \quad y = y_0 + bt$

Direction: $\langle a, b \rangle$, passes through $(x_0, y_0)$

Parametric Circle

$x = h + r\cos t, \quad y = k + r\sin t, \quad 0 \leq t \leq 2\pi$

Polar Coordinates

Point $(r, \theta)$ where:

• $r$ = distance from origin
• $\theta$ = angle from positive x-axis

Conversion

$x = r\cos\theta, \quad y = r\sin\theta$ $r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$

Vectors

Definition

$\vec{v} = \langle a, b \rangle$

Magnitude

$|\vec{v}| = \sqrt{a^2 + b^2}$

Unit Vector

$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$

Operations

$\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$ $c\vec{v} = \langle cv_1, cv_2 \rangle$

Dot Product

$\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 = |\vec{u}||\vec{v}|\cos\theta$

Matrices (Introduction)

2×2 Matrix

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

Matrix Multiplication

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$

Transformation Matrices

Rotation by $\theta$: $R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

AP Precalculus Tip: Unit 4 topics (parametric, polar, vectors, matrices) are emphasized for understanding how functions can be represented beyond $y = f(x)$.