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Precalculus Unit 11: Conics, parametrics, vectors & matrices

This unit extends functions into parametric equations, vectors and matrices, which set up multivariable calculus and linear algebra. (It isn't on the AP Precalculus exam.)

Covers: Conic sections, parametric equations, vectors, matrix operations & transformations

Study guide

Conic sections are curves formed by slicing a cone. A parabola (y = ax² or x = ay²) has a focus and directrix. An ellipse, x²/a² + y²/b² = 1, is a stretched circle with two foci; if a = b, it's a circle. A hyperbola, x²/a² − y²/b² = 1, has two branches that approach asymptotes. Shifted conics replace x and y with (x − h) and (y − k), and completing the square puts an equation into standard form.

Parametric equations give x and y as separate functions of a parameter t, like x = cos t, y = sin t. They describe motion: t is often time, and the graph shows the path with direction. Eliminating the parameter (solving one equation for t and substituting into the other) gives a rectangular equation for the path.

Vectors have magnitude and direction. A vector ⟨a, b⟩ has magnitude √(a² + b²). Add vectors component by component, multiply by scalars to stretch them, and use the dot product to find angles.

Matrices are rectangular arrays. Add matrices entry by entry; multiply rows by columns (so AB ≠ BA in general). The determinant of [a b; c d] is ad − bc, and a 2×2 matrix is invertible exactly when its determinant isn't zero. Multiplying a point's coordinates by a 2×2 matrix performs a linear transformation such as a rotation, reflection or dilation, and |det| gives the area scale factor.

For an ellipse x²/a² + y²/b² = 1 with a > b, the vertices are (±a, 0), the major and minor axes have lengths 2a and 2b, and the foci are (±c, 0) with c² = a² − b². For a hyperbola x²/a² − y²/b² = 1, c² = a² + b² and the asymptotes are y = ±(b/a)x. A vector with magnitude |v| and direction angle θ (measured from the positive x-axis) has components ⟨|v|cos θ, |v|sin θ⟩. To add vectors given by speed and heading, convert each to components, add, then convert back using magnitude √(a² + b²) and angle tan⁻¹(b/a), adjusted for the quadrant. Compass bearings are measured clockwise from north, so convert them to standard angles first.

The dot product is ⟨a, b⟩·⟨c, d⟩ = ac + bd = |u||v|cos θ, so cos θ = (u·v)/(|u||v|). Two vectors are perpendicular exactly when their dot product is 0, and parallel when one is a scalar multiple of the other. An m×n matrix can multiply an n×p matrix only when the inner dimensions match, and the product is m×p. The entry in row i, column j is row i of the first matrix dotted with column j of the second.

The segment from (x₁, y₁) to (x₂, y₂) is traced by x = x₁ + (x₂ − x₁)t and y = y₁ + (y₂ − y₁)t for 0 ≤ t ≤ 1. At t = 0 you are at the start, and at t = 1 you are at the end.

Key ideas

• Parametric equations give x and y as separate functions of a parameter t.
• Eliminate the parameter to get a rectangular equation.
• A vector has magnitude and direction; add vectors component by component.
• Matrices multiply row by column; the product AB requires columns of A = rows of B.
• A 2×2 matrix is invertible exactly when its determinant isn't zero.
• Conics: circles, ellipses, parabolas and hyperbolas each have a standard equation.

Formulas and rules

Common mistakes

• Assuming matrix multiplication is commutative (AB ≠ BA in general).
• Forgetting the direction of motion when sketching parametric curves.

Words to know

Magnitude of a vector

|⟨a, b⟩| = √(a² + b²)

Determinant of [a b; c d]

ad − bc

Invertible 2×2 matrix

A matrix whose determinant is not zero.

Matrix multiplication order

Not commutative: AB ≠ BA in general.

Eliminating the parameter

Solve one equation for t and substitute into the other.

Ellipse (standard form)

x²/a² + y²/b² = 1

Hyperbola (standard form)

x²/a² − y²/b² = 1

Parabola (conic)

Points equidistant from a focus and a directrix.

Parametric equations

x and y given as functions of a parameter t.

Vector

A quantity with magnitude and direction, written ⟨a, b⟩.

Dot product

⟨a, b⟩·⟨c, d⟩ = ac + bd

Matrix transformation

Multiplying coordinates by a matrix to rotate, reflect or dilate.

Area scale factor

|det A| for the transformation by matrix A.

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.