Unit Tangent Vector Calculator

This unit tangent vector calculator finds T(t) = r′(t)/|r′(t)| for a curve r(t) = ⟨x(t), y(t), z(t)⟩ whose components are polynomials in t. It differentiates each component, finds the length of r′(t), and gives T at the value of t you choose exactly — for example ⟨1/3, 2/3, 2/3⟩ — with decimals when roots appear.

Formula

\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\lVert \mathbf{r}'(t) \rVert}

Divide the velocity vector by its length so the result has length 1.

\lVert \mathbf{r}'(t) \rVert = \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2}

The length (speed) of r′(t).

How to use this calculator

Choose a 2D or 3D curve.
Type each component as a polynomial in t, such as t, t^2 and 2t^3/3.
Enter the value of t and press Calculate: you get T at that point, the speed |r′(t)|, and the general formula for T(t).

Worked examples

Easy: a 2D curve at t = 0

r′(t) = ⟨3, 4 − 2t⟩, so r′(0) = ⟨3, 4⟩.

|r′(0)| = √(9 + 16) = 5.

T(0) = ⟨3, 4⟩/5 = ⟨3/5, 4/5⟩.

Medium: a 3D curve with a whole-number speed

r′(t) = ⟨1, 2t, 2t²⟩, so r′(1) = ⟨1, 2, 2⟩.

|r′(1)| = √(1 + 4 + 4) = 3.

T(1) = ⟨1/3, 2/3, 2/3⟩.

Hard: a surd in the length

r′(t) = ⟨2t, 3t²⟩, so r′(1) = ⟨2, 3⟩.

|r′(1)| = √(4 + 9) = √13.

T(1) = ⟨2/√13, 3/√13⟩ = ⟨(2√13)/13, (3√13)/13⟩ ≈ ⟨0.5547, 0.8321⟩.

Common mistakes

Dividing r(t) by its length instead of r′(t): the tangent comes from the derivative.
Forgetting to square each component before adding under the square root.
Trying to find T where r′(t) = 0: the curve has no defined direction there.
Leaving out the z-component of a 3D curve when finding the length.

Frequently asked questions

What is the unit tangent vector?

T(t) is the direction of motion along a curve r(t), scaled to length 1: T(t) = r′(t)/|r′(t)|. It points the way the curve is heading at each t.

How do you find the unit tangent vector at a point?

Differentiate each component to get r′(t), substitute the value of t, find the length of that vector, and divide each component by the length.

Can I enter sin t or e^t?

Not yet — each component must be a polynomial in t, such as 3t^2 - t + 1. The answer is then exact.

How is the answer checked?

The calculator squares the components of T and adds them: the total must be exactly 1.

Result