Vector Functions and Space Curves

A vector function assigns a vector to each value of a parameter; as the parameter runs, its tip sweeps out a curve through space. Each component is an ordinary scalar function, so limits, derivatives, and integrals all act componentwise, and single-variable calculus carries over with no new machinery. The one genuinely new object is the derivative of a vector function, the tangent vector to the curve it traces.

Vector functions and space curves

The domain is the set of $t$ for which every component is defined. For $r (t) = ⟨ t^{3}, ln (3 - t), t ⟩$ , the logarithm needs $t < 3$ and the root needs $t \geq 0$ , so the domain is $[0, 3)$ .

Limits and continuity pass through the components untouched.

A continuous vector function is the same data as a curve. As $t$ ranges over an interval, the point $P (f (t), g (t), h (t))$ moves through space, and the set of all such points is a space curve.

A linear vector function reproduces a line: $r (t) = ⟨ 1 + t, 2 + 5 t, - 1 + 6 t ⟩$ is the line through $(1, 2, - 1)$ with direction $⟨ 1, 5, 6 ⟩$ . Mixing trigonometric and linear components produces the archetypal space curve, the helix. For $r (t) = cos t \overset{}{^} + sin t \overset{}{^} + t \hat{k}$ , the first two components satisfy $x^{2} + y^{2} = 1$ , so the curve lies on the cylinder of radius $1$ , while $z = t$ raises it steadily as it circles: a corkscrew, the shape of a spring and of the DNA backbone.

The helix r(t) = (cos t, sin t, t) circles the unit cylinder while rising steadily, so it winds upward one full turn every 2 pi in t.

Vector functions also parametrize curves of intersection of surfaces.

Derivatives

The derivative of a vector function copies the single-variable definition, with a difference quotient of vectors.

The geometry is in the difference quotient. The vector $r (t + h) - r (t)$ is a secant joining two nearby points on the curve; dividing by $h$ scales it without changing direction, and as $h \to 0$ the secant swings onto the tangent line.

The secant vector r(t+h) - r(t) joins two nearby points; scaled by 1/h and taken to the limit h -> 0, it becomes the tangent vector r'(t).

Rather than take the limit directly, differentiate each component.

For $r (t) = (1 + t^{3}) \overset{}{^} + t e^{- t} \overset{}{^} + sin 2 t \hat{k}$ , differentiating each component gives $r^{'} (t) = 3 t^{2} \overset{}{^} + (1 - t) e^{- t} \overset{}{^} + 2 cos 2 t \hat{k}$ .

Dividing the tangent vector by its length removes the speed information and keeps only the direction of travel.

The unit tangent vector T(t) at several points along a curve, each pointing in the direction of increasing t and each of length one.

The second derivative $r^{''} (t) = (r^{'} (t))^{'}$ is again taken componentwise; for the helix above, $r^{''} (t) = ⟨ - 2 cos t, - sin t, 0 ⟩$ . Read as the motion of a particle, $r^{'}$ is its velocity and $r^{''}$ its acceleration.

Differentiation rules

Every product rule of scalar calculus has a vector counterpart, one for each way two vector functions can be combined.

Order matters in the cross-product rule, since the cross product is anticommutative. One consequence: a curve confined to a sphere has its tangent vector perpendicular to its position vector.

A constant length means $r (t) \cdot r (t) = ∥ r (t) ∥^{2}$ is constant, so differentiating with the dot-product rule gives $2 r^{'} (t) \cdot r (t) = 0$ , hence $r^{'} ⊥ r$ . Geometrically, a curve on a sphere centered at the origin always has its tangent perpendicular to its radius.

Integrals

Integration is componentwise too, so the antiderivative of a vector function is the antiderivative of each component, and the Fundamental Theorem of Calculus extends verbatim.

An indefinite integral carries a vector constant of integration. For $r (t) = 2 cos t \overset{}{^} + sin t \overset{}{^} + 2 t \hat{k}$ ,

\int r (t) d t = 2 sin t \overset{}{^} - cos t \overset{}{^} + t^{2} \hat{k} + C .

Integration recovers position from a known velocity, and velocity from a known acceleration, exactly as in one dimension. Applied to Newton's second law, $F = m a$ , it turns a force field into a trajectory.

Each operation applies the scalar version to every component and reassembles the result.

Operation	Vector form	How it is computed
Limit	$lim_{t \to a} r (t)$	limit of each component
Continuity	$lim_{t \to a} r (t) = r (a)$	every component continuous
Derivative	$r^{'} (t)$	derivative of each component
Integral	$\int_{a}^{b} r (t) d t$	integral of each component

The only genuinely vector-valued subtleties are the product rules, where the dot and cross products each contribute their own version, and the geometric reading of $r^{'}$ as a tangent vector — neither of which has a one-dimensional analog.

A curve can also be studied through its projections onto the coordinate planes. Dropping the $z$ -coordinate of $r (t)$ leaves its shadow on the $x y$ -plane, and comparing the three shadows often clarifies a curve that is hard to read in perspective.

A space curve and its shadow on the xy-plane; the projection drops the z-coordinate, turning the curve into a plane curve on the floor.