AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.9 — Vector-Valued Functions
Notes — Position Vectors that Depend on a Parameter
💡 Learning Objectives (4.9.A)
By the end of this lesson you will be able to:
- Define a vector-valued function as r(t) = ⟨x(t), y(t)⟩
- Connect vector-valued functions to parametric equations
- Compute the position vector at specific values of t
- Evaluate magnitude, direction, and rate-of-change of the position vector
1. From Parametric to Vector
A parametric function (x(t), y(t)) gives the COORDINATES of a moving point. We can package the same information into a single vector function:
r(t) = ⟨x(t), y(t)⟩ or equivalently r(t) = x(t) i + y(t) j
This is a VECTOR-VALUED FUNCTION: input t, output a vector. The vector r(t) is the POSITION VECTOR of the point at time t — an arrow from the origin to (x(t), y(t)).
2. Evaluating a Vector-Valued Function
📘 Example — Evaluate r(t)
Let r(t) = ⟨t² − 1, 3t⟩.
- r(0) = ⟨−1, 0⟩
- r(1) = ⟨0, 3⟩
- r(2) = ⟨3, 6⟩
- r(−1) = ⟨0, −3⟩
3. Magnitude as a Function of t
The DISTANCE FROM THE ORIGIN at time t is the magnitude of r(t):
|r(t)| = √(x(t)² + y(t)²)
📘 Example — Distance from origin
r(t) = ⟨3 cos t, 3 sin t⟩.
- |r(t)| = √(9 cos²t + 9 sin²t) = √9 = 3 for every t
- So the particle is always at distance 3 from the origin — i.e. on a circle of radius 3.
4. Sketching the Path
The path of r(t) is exactly the parametric curve (x(t), y(t)). Sketch it the same way: tabulate t-values, plot the resulting points, connect with a smooth curve, indicate direction.
5. Vector Operations on Vector Functions
Vector-valued functions can be added, subtracted, and scaled like ordinary vectors:
- (r₁ + r₂)(t) = r₁(t) + r₂(t) — componentwise sum
- (k · r)(t) = ⟨k · x(t), k · y(t)⟩
- (r₁ − r₂)(t) = ⟨x₁(t) − x₂(t), y₁(t) − y₂(t)⟩ — note: this gives a vector from particle 2 to particle 1 at time t
6. Average Rate of Change of r(t)
The AVERAGE RATE OF CHANGE of a vector function over [t₁, t₂] is:
AROC = (r(t₂) − r(t₁)) / (t₂ − t₁) = ⟨(x(t₂) − x(t₁))/(t₂−t₁), (y(t₂) − y(t₁))/(t₂−t₁)⟩
This is itself a VECTOR — its components are the average rates of change of x and y respectively. Geometrically, it points from r(t₁) to r(t₂) divided by the time interval, i.e. it represents AVERAGE VELOCITY.
📘 Example — Average velocity vector
r(t) = ⟨t², 2t⟩ on [1, 3].
- r(1) = ⟨1, 2⟩, r(3) = ⟨9, 6⟩
- Δr = ⟨8, 4⟩, Δt = 2
- AROC = ⟨4, 2⟩ — average velocity vector
- Magnitude: √20 ≈ 4.47 — the average SPEED over [1, 3]
7. Direction of Motion at an Instant
Although AP precalculus does not formally use derivatives, you can describe the direction of motion at a given t qualitatively from the AROC:
- If AROC of x is positive over an interval that contains t: motion is to the right
- If AROC of y is positive over an interval that contains t: motion is upward
- Picking t₂ very close to t₁ approximates the instantaneous direction
8. Position Vector vs. Displacement Vector
Two different vectors get used in this topic:
- Position vector r(t): from origin to current location
- Displacement vector Δr = r(t₂) − r(t₁): from one position to a later one
The displacement is the SHORTCUT of the path between the two times, NOT the path length. The path length is generally LONGER (it might curve back and forth), but the displacement is a straight arrow.
9. Summary
- A vector-valued function r(t) = ⟨x(t), y(t)⟩ describes a moving point as a position vector
- Magnitude |r(t)| is the distance from the origin at time t
- Vector functions add, subtract, and scale componentwise
- AROC over an interval is itself a vector — average velocity
- Position vector ≠ displacement vector; both are useful in motion problems