A vector-valued function uses one input, usually time t, to produce a vector output such as r(t) = <x(t), y(t), z(t)>. As t changes, the tip of the position vector moves through space and traces a curve. This idea is essential for describing motion in physics, robotics, computer graphics, and orbital paths.
Instead of studying one coordinate at a time, you study a full geometric path with direction and position built in.
The main calculus rules work component by component, so limits, derivatives, and integrals of r(t) are found by applying ordinary calculus to x(t), y(t), and z(t). The derivative r'(t) gives the velocity vector tangent to the curve, while the second derivative r''(t) gives acceleration. The integral of a vector-valued function can represent accumulated displacement or total change in position.
These tools connect algebraic formulas to visual features such as motion direction, speed, curvature, and tangent lines.
Key Facts
- A vector-valued function in 3D has the form r(t) = <x(t), y(t), z(t)>.
- The space curve is the set of points traced by the tip of r(t) as t varies.
- lim t->a r(t) = <lim t->a x(t), lim t->a y(t), lim t->a z(t)> when all component limits exist.
- r'(t) = <x'(t), y'(t), z'(t)> gives the velocity vector and is tangent to the curve.
- Speed is the magnitude of velocity: speed = |r'(t)| = sqrt((x'(t))^2 + (y'(t))^2 + (z'(t))^2).
- Integral from a to b of r(t) dt = <integral from a to b x(t) dt, integral from a to b y(t) dt, integral from a to b z(t) dt>.
Vocabulary
- Vector-valued function
- A function that assigns a vector to each input value, often written as r(t) = <x(t), y(t), z(t)>.
- Position vector
- A vector drawn from the origin to the point on the curve corresponding to a particular value of t.
- Space curve
- A curve in three-dimensional space traced by the endpoint of a vector-valued function.
- Velocity vector
- The derivative r'(t), which gives the instantaneous direction and rate of motion along the curve.
- Acceleration vector
- The second derivative r''(t), which describes how the velocity vector changes over time.
Common Mistakes to Avoid
- Treating r(t) as a scalar function, which is wrong because r(t) has multiple components and must be handled as a vector.
- Finding the derivative by differentiating only one component, which is wrong because r'(t) requires differentiating every component separately.
- Confusing velocity with speed, which is wrong because velocity is a vector while speed is the scalar magnitude |r'(t)|.
- Assuming the position vector and tangent vector point in the same direction, which is wrong because the position vector points from the origin to the curve while the tangent vector points along the curve.
Practice Questions
- 1 For r(t) = <t^2, 3t, sin t>, find r'(t) and r''(t).
- 2 For r(t) = <cos t, sin t, 2t>, find the velocity vector and speed at t = pi/2.
- 3 A particle has position r(t) = <t, t^2, 0>. Explain why the position vector and velocity vector are generally not parallel, and identify any value of t where they are parallel.