Unit Tangent Vectors for Non-Unit-Speed Curves

From Velocity to Unit Tangent

A parametrized curve comes with a velocity vector: the derivative of its position vector. That velocity carries two kinds of information at once. Its direction indicates the curve’s instantaneous direction of travel, while its magnitude tells how quickly the parameterized point is moving.

For some questions, we need the direction without the speed. We obtain it by dividing the velocity by its length. The resulting vector has length one and points along the curve in the direction of increasing parameter.

Regularity matters: if the velocity is zero, the formula would require division by zero, so it does not define a unit tangent there. A curve may be regular at some parameter values but not at others.

Why Normalizing the Velocity Gives the Tangent Direction

The derivative \(\mathbf{r}'(t)\) describes the limiting direction of nearby secants to the curve. To see this, take a small positive increment \(h\). The vector from \(\mathbf{r}(t)\) to \(\mathbf{r}(t+h)\), divided by its length, is a unit secant direction:

For \(h>0\), the numerator can be written as \(h\) times the difference quotient, and the same factor \(h\) cancels from the numerator and denominator. Thus this unit secant direction equals

As \(h\) approaches zero, the difference quotient approaches \(\mathbf{r}'(t)\). If \(\mathbf{r}'(t)\ne\mathbf{0}\), its length approaches the positive number \(\|\mathbf{r}'(t)\|\). Therefore, the unit secant direction approaches \(\mathbf{r}'(t)/\|\mathbf{r}'(t)\|\). This proves that the formula gives the limiting tangent direction for increasing parameter values.

A Reliable Calculation

For a curve given by \(\mathbf{r}(t)\), calculate its unit tangent in three steps: differentiate each component, find the length of the resulting velocity vector, and divide every velocity component by that length. Keep the speed in the denominator until all components have been divided by it.

Variable Speed Does Not Change the Normalization Rule

The speed need not be constant. In that case, it remains inside the formula as a function of the parameter. The unit tangent can change direction along the curve even though its length stays fixed at one.

Orientation and Regularity

The unit tangent describes the direction of increasing parameter. If a curve is traversed in reverse, its velocity points the opposite way, and so does its unit tangent. For example, if a new parameter \(u\) is related to \(t\) by a differentiable function \(t=t(u)\), the chain rule gives the new velocity as \(\mathbf{r}'(t(u))\,dt/du\). When \(dt/du\ne 0\), normalizing this vector preserves the original unit tangent if \(dt/du>0\), and reverses it if \(dt/du<0\). The tangent line is the same, but its chosen direction depends on the orientation of the parameter.

Regularity is a local condition on the velocity: at the parameter value in question, the derivative must be nonzero. If it is zero, do not apply the unit tangent formula there by cancellation or by assuming the curve has an ordinary tangent direction. First recognize that this particular formula is undefined at that value.

Common Mistake: Treating Velocity as a Unit Tangent

A frequent error is to differentiate the curve and stop, calling the result the unit tangent. For the helix in the first example, this would give \(\langle -3\sin t,3\cos t,4\rangle\). At \(t=0\), that vector is \(\langle 0,3,4\rangle\), whose length is \(5\), not \(1\). It is the velocity, not the unit tangent.

The correction is to divide by the speed, which is \(5\) in this example:

A related error is to divide by the position vector’s length \(\|\mathbf{r}(t)\|\) instead of the velocity’s length \(\|\mathbf{r}'(t)\|\). The unit tangent is defined from velocity. The position vector points from the origin to the curve point; in general, it does not give the curve’s tangent direction.

Why the Unit Tangent Is Useful

Velocity combines direction and rate of motion, and both are useful in motion problems. But when the question concerns the curve’s direction at a point, speed is irrelevant. The unit tangent provides a consistent direction vector even when the parameter moves quickly in one part of the curve and slowly in another.

It is also an essential ingredient in later descriptions of how a curve’s direction changes. For now, the central calculation is simple: differentiate, compute the speed, and divide. The regularity check ensures the denominator is nonzero, and the length check confirms that the result is genuinely a unit vector.