4.7 Vector-Valued Functions

What is a Vector-Valued Function?

A vector-valued function assigns a vector to each value of a parameter, often time *t*, describing motion in the plane: v(t) = ⟨x(t), y(t)⟩. As *t* changes, the vector's terminal point traces a parametric curve, modeling position, velocity, or displacement over time.

Modeling Motion with Vectors

The position vector ⟨x(t), y(t)⟩ locates an object at time *t*. Comparing positions at two times gives a displacement vector: Δv = ⟨x(t₂)−x(t₁), y(t₂)−y(t₁)⟩. Dividing displacement by elapsed time gives average velocity, a vector showing both speed and direction of travel between two instants.

Combining and Scaling Vectors

Vector-valued functions can be added, subtracted, or scaled component-wise, just like vectors. Adding two vector functions combines their effects (e.g., wind plus plane velocity), while scaling changes magnitude without changing direction (unless the scalar is negative, which reverses it).

Key terms

Vector-valued function

A function v(t) = ⟨x(t), y(t)⟩ that assigns a vector to each parameter value, often modeling motion.

Position vector

A vector ⟨x(t), y(t)⟩ that gives the location of a point at time t.

Displacement vector

The difference between two position vectors: ⟨x(t₂)−x(t₁), y(t₂)−y(t₁)⟩.

Average velocity vector

Displacement vector divided by the elapsed time, showing direction and rate of motion.

Adding vector-valued functions

Add corresponding components: ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩.

Try an AP-style question

r(t) = ⟨t, t²⟩. What is r(3)?

1. ⟨9, 3⟩
2. ⟨3, 6⟩
3. 12
4. ⟨3, 9⟩