Functions, Vectors, Matrices — AP Precalculus
1. Parametric Equations: Eliminating the Parameter ★★☆☆☆ ⏱ 4 min
Parametric equations define coordinate values as separate functions of a shared independent variable, called the parameter (almost always denoted $t$, representing time in most motion problems). For 2D curves, they are written as $x = f(t), \quad y = g(t)$. Eliminating the parameter rewrites these two equations as a single Cartesian relation in terms of $x$ and $y$ only.
For trigonometric parametric equations, use the Pythagorean identity $\cos^2 t + \sin^2 t =1$ to eliminate $t$ without direct substitution. For example, $x = 4\cos t$, $y =4\sin t$ gives $x^2 + y^2 =16$, a circle of radius 4. If $t \in [0, \pi]$, restrict the range to $y \geq 0$ to get the upper semicircle.
Exam tip: Always add explicit domain and range restrictions after eliminating the parameter — the parametric curve is only a subset of the full Cartesian graph, and missing restrictions will cost you points.
2. Vectors in 2D and 3D ★★☆☆☆ ⏱ 3 min
To find the vector from point $A(x_1, y_1, z_1)$ to point $B(x_2, y_2, z_2)$, subtract the initial point components from the terminal point components:
\vec{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle
3. Core Vector Operations: Magnitude, Sum, Dot Product ★★★☆☆ ⏱ 4 min
All core vector operations tested on AP Precalculus are calculated component-wise, with clear geometric interpretations:
- **Magnitude**: The length of the vector, calculated by the extended Pythagorean theorem: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
- **Vector addition**: Add corresponding components of the two vectors, matching the geometric tip-to-tail rule: $\mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y, u_z + v_z \rangle$
- **Dot product (scalar product)**: Returns a scalar equal to the sum of products of corresponding components, used to find the angle between two vectors: $\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z = |\mathbf{u}| |\mathbf{v}| \cos\theta$. Non-zero vectors are perpendicular if and only if their dot product is 0.
4. Matrices: Operations and Linear Transformations ★★★☆☆ ⏱ 5 min
Core matrix operations follow specific rules, all tested on the AP Precalculus exam:
- **Addition**: Only possible for matrices with identical dimensions. Add corresponding entries.
- **Scalar multiplication**: Multiply every entry in the matrix by the constant scalar.
- **Matrix multiplication**: Multiply $A_{m \times n}$ by $B_{n \times p}$ to get $C_{m \times p}$, where each entry $c_{ij}$ is the dot product of row $i$ of $A$ and column $j$ of $B$. Matrix multiplication is not commutative, so $AB \neq BA$ in most cases.
To apply a transformation, multiply the transformation matrix by the column vector of the point (matrix first, vector second). For a sequence of transformations, multiply matrices in reverse order of application: the first transformation applied goes on the right side of the product.
5. AP-Style Concept Check Practice ★★★☆☆ ⏱ 4 min
Common Pitfalls
Why: You focus only on substitution algebra and ignore original $t$ limits; the full Cartesian graph is larger than the parametric curve
Why: Order of subtraction is easily mixed up
Why: You confuse dot product with regular vector multiplication
Why: You assume matrix multiplication is commutative like regular scalar multiplication
Why: You mix up clockwise and counterclockwise rotation rules