# What is the unit tangent vector T(t)?

Finds the unit tangent vector T(t) = r′(t)/‖r′(t)‖ of a curve r(t) in 2D or 3D, and its value at t = a.

- Page: https://www.acalculator.org/math/unit-tangent-vector-calculator
- JSON spec: https://www.acalculator.org/math/unit-tangent-vector-calculator.json
- Version: 0f4f93444b64

## Default answer

Example with the default inputs (x(t) 3t^2 + 2t, y(t) 2 - 4t^3, z(t) (empty for a plane curve) 6t + 5, At t = a 1): The unit tangent vector at t = 1 is (4/sqrt(61), -6/sqrt(61), 3/sqrt(61)).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | x(t) | The x component of r(t), a function of t. |
| y | y(t) | The y component of r(t), a function of t. |
| z | z(t) (empty for a plane curve) | The z component of r(t), a function of t. |
| a | At t = a | Where to evaluate T: a number or a constant such as pi/2. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| at | T(a) | The unit tangent vector at t = a. |
| formula | T(t) = | r′(t) divided by its length ‖r′(t)‖. |
| derivative | r′(t) = | The tangent vector, each component differentiated. |
| speed | ‖r′(a)‖ | The length of r′(a), the speed at t = a. |

## Method

T(t) = r′(t) / ‖r′(t)‖, each component of r′ from a computer algebra system, checked numerically.

## Assumptions

- The variable is t; angles are in radians.
- T(a) needs r′(a) ≠ 0.

## Worked examples

1. x = 3t^2 + 2t, y = 2 - 4t^3, z = 6t + 5, a = 1 gives derivative = (6t + 2, -12t^2, 6), formula = (6t + 2, -12t^2, 6)/sqrt(144t^4 + 36t^2 + 24t + 40), at = (4/sqrt(61), -6/sqrt(61), 3/sqrt(61)), speed = 15.620499. Source: OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions, Example 3.7 (https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions), part b: u′(t) = (6t + 2, −12t², 6).
2. x = cos(t), y = sin(t), a = pi/2 gives derivative = (-sin(t), cos(t)), formula = (-sin(t), cos(t)), at = (-1, 0), speed = 1. Source: OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions, Example 3.7 (https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions), part a: T(t) = −sin t i + cos t j.
3. x = t^2 - 3, y = 2t + 1, z = t - 2, a = 2 gives at = (4/sqrt(21), 2/sqrt(21), 1/sqrt(21)), speed = 4.582576.

## FAQ

### What is the unit tangent vector?

For a smooth curve r(t), the unit tangent vector is T(t) = r′(t)/‖r′(t)‖. It points the way the curve moves as t grows, and its length is 1. It exists wherever r′(t) is not the zero vector.

### How do I find the unit tangent vector?

Differentiate each component to get r′(t). Find its length ‖r′(t)‖ = √(x′² + y′² + z′²). Divide each component of r′(t) by that length. For r(t) = (cos t, sin t), r′(t) = (−sin t, cos t) has length 1, so T(t) = (−sin t, cos t).

### Why is T not defined at some points?

Where r′(t) = 0, the length is 0 and the division is not possible. The curve may have a sharp point there: r(t) = (t³, t²) has r′(0) = (0, 0) and a cusp at the origin. The page says so instead of giving a vector.

### Does the unit tangent vector depend on the parametrization?

Its direction depends only on the direction of travel. A new parameter that runs the same way (t = 2s) gives the same T at each point; one that runs the other way (t = −s) gives −T.

### How does T relate to curvature?

The curvature is κ = ‖T′(t)‖/‖r′(t)‖: how fast the unit tangent turns per unit of arc length. The principal unit normal is N(t) = T′(t)/‖T′(t)‖.

### How do I enter a plane curve?

Leave z(t) empty. The page then works with r(t) = (x(t), y(t)), and the answers have two components.

### How is the answer checked?

Each derivative comes from a computer algebra system and must match a numeric difference quotient. The exact value of T(a) is shown only when each of its components equals the decimal value to 10⁻⁹; otherwise the page shows decimals to 10 significant figures.

## Sources

- OpenStax, Calculus Volume 3, section 3.2 Calculus of Vector-Valued Functions (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-3/pages/3-2-calculus-of-vector-valued-functions
- OpenStax, Calculus Volume 3, section 3.3 Arc Length and Curvature (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature