Unit 4: Functions Involving Parameters, Vectors & Matrices

Vector-Valued Functions

Representing and operating on vectors and vector-valued functions.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Parametric Curve's Position, as a Single Object

A parametric curve's (x(t), y(t)) pair already tracks a moving point's position at every instant t. Before reading on: what would it mean to treat that whole pair — not just its two separate coordinate functions — as one mathematical object with its own magnitude and direction?

Definition — Vector and Vector-Valued Function

A vector v = ⟨vₓ, vy⟩ has both a magnitude, |v| = √(vₓ² + vy²), and a direction. A vector-valued function r(t) = ⟨x(t), y(t)⟩ assigns a vector — typically a position — to every value of the parameter t, the same pairing a parametric curve already uses, now carrying magnitude and direction as first-class properties.

|v| = \sqrt{v_x^2 + v_y^2}

Worked Example — Vector Operations

Let u = ⟨3, −4⟩ and v = ⟨−1, 2⟩. Sum: u + v = ⟨3 + (−1), −4 + 2⟩ = ⟨2, −2⟩. Scalar multiple: 2u = ⟨6, −8⟩. Magnitude: |u| = √(3² + (−4)²) = √25 = 5. Direction angle: since u sits in Quadrant IV (positive x, negative y), the reference angle is arctan(4/3) ≈ 53.1°, so u's direction is about −53.1° (equivalently 306.9°).

Worked Example — Average Rate of Change of a Vector-Valued Function

For r(t) = ⟨t², 2t⟩, find the average rate of change from t = 1 to t = 3. r(1) = ⟨1, 2⟩ and r(3) = ⟨9, 6⟩. The displacement is r(3) − r(1) = ⟨8, 4⟩, so the average rate of change is ⟨8, 4⟩/(3 − 1) = ⟨4, 2⟩ — a vector, not a single number, since position itself is a vector. Its magnitude, |⟨4, 2⟩| = √(16 + 4) = √20 = 2√5 ≈ 4.47, gives the average speed; its direction, arctan(2/4) ≈ 26.6°, gives the average heading over that interval.

Tip

Average rate of change of a vector-valued function works exactly like the scalar version from Unit 1 — displacement divided by elapsed parameter — just applied component-by-component, since subtracting and dividing vectors already work componentwise.

Common Mistakes

Adding two vectors' magnitudes instead of adding their components.
|u| + |v| is not |u + v| in general — combine vectors by adding matching components first (⟨2,−2⟩ above), and only take a magnitude afterward if a magnitude is actually what's needed.
Finding a direction angle with arctan alone and skipping the quadrant check.
arctan(vy/vₓ) only returns a reference angle in (−90°, 90°) — as with polar coordinates, the vector's actual quadrant (from the signs of vₓ and vy) determines whether that angle needs adjusting.

Key Takeaways

A vector ⟨vₓ, vy⟩ has magnitude √(vₓ² + vy²) and a direction angle found from its components, with the correct quadrant.
A vector-valued function r(t) = ⟨x(t), y(t)⟩ generalizes a parametric curve, treating position as a single vector quantity.
Average rate of change of a vector-valued function is the displacement vector divided by the change in t — a vector result, with its own magnitude and direction.

Summary

Vectors describe position, displacement, and direction as single objects. The final lesson of this course studies matrices — mathematical objects that act on vectors and points as functions in their own right, transforming the entire plane at once.

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