A vector-valued function takes one number t and returns a point in space: r(t) = ⟨x(t), y(t), z(t)⟩. As t runs, the point traces a curve.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The parameter is usually time
Think of t as a clock and r(t) as where a particle is at that moment. The curve is the trail it leaves.
A circle
r(t) = ⟨cos t, sin t, 0⟩ traces the unit circle once as t runs from 0 to 2π. The identity cos²t + sin²t = 1 is why.
A helix
Add a rising third component: ⟨cos t, sin t, t⟩ circles while climbing. That is a spring, a spiral staircase or a strand of DNA.
Path and parametrisation differ
Two different functions can trace the same curve at different speeds. The path is the picture; the parametrisation is the schedule.
The domain
Every component function must be defined. The domain of r is the overlap of the three separate domains.
One input, several outputs
r(t) = ⟨x(t), y(t), z(t)⟩ gives a position in space for each value of t, tracing a curve. It is the parametric idea from precalculus with a third component added.
Calculus applies componentwise
Limits, derivatives and integrals of a vector function are computed on each component separately. That reduces new machinery to the single-variable calculus already available.
The domain is the intersection
The function is defined only where every component is defined. One component with a restricted domain restricts the whole curve, which is easy to overlook.
Direction is part of the curve
Increasing t traces the curve in a definite direction. The same set of points traversed the other way is a different parametrisation, and orientation matters for line integrals later.
Step 2: Try It Yourself
Tap and try it out.
- Angle60° = π/3 rad
- x-coordinate0.500
- y-coordinate0.866
- cos 60°0.500
- sin 60°0.866
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Marcus Locate a Particle
Marcus evaluates r(t) = ⟨2t, t², 5⟩ at t = 3.
- Step 1
He substitutes into the first component: 2 × 3 = 6.
Step 4: Your Turn
Practice makes it stick.
The Spring
Problem 1 of 2
r(t) = ⟨cos t, sin t, 2t⟩. What is the z-coordinate at t = 3?
The Start
Problem 2 of 2
r(t) = ⟨t + 1, 3t, t³⟩. What is the x-coordinate at t = 0?
Trace the Curve
1 of 8
r(t) = ⟨3t, t², 1⟩ at t = 2. What is the x-coordinate?
2 of 8
r(t) = ⟨3t, t², 1⟩ at t = 2. What is the y-coordinate?
3 of 8
r(t) = ⟨cos t, sin t, 0⟩. What is the distance from the origin at any t?
4 of 8
How many full circles does ⟨cos t, sin t, 0⟩ trace as t runs from 0 to 4π?
5 of 8
r(t) = ⟨t, t, t⟩. Is this a straight line? 1 yes, 0 no.
6 of 8
r(t) = ⟨cos t, sin t, 4t⟩. How much does z rise over one full turn, using 2π as about 6.28?
7 of 8
Order the values of t so the particle on ⟨t, t², 0⟩ visits them in sequence.
- 1t = 0
- 2t = 1
- 3t = 2
- 4t = −1
8 of 8
r(t) = ⟨5, t, 0⟩. What is the x-coordinate at t = 100?
Step 5: Quick Check
Show what you know.
Question 1 of 2
r(t) = ⟨t², 2t, 0⟩ at t = 4. What is the y-coordinate?
Question 2 of 2
What does the parameter t usually represent?
What You Learned
- A vector-valued function turns one number into a point, tracing a curve.
- ⟨cos t, sin t, 0⟩ is a circle; adding a rising z makes it a helix.
- The same path can be traced by different parametrisations at different speeds.