Unit Tangent Vector Calculator
Compute the unit tangent vector for a curve r(t). Enter x(t), y(t), z(t) and point t to get T(t) instantly.
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Helping Notes
Only four inputs are required: x(t), y(t), z(t), and the numeric point t. Trig names are sin, cos, tan; use ^ for powers.
If ‖r′(t)‖ = 0 at your point, T(t) is undefined. Adjust the point or functions so the derivative is nonzero.
Results
r′(t)
‖r′(t)‖
T(t)
What Is a Unit Tangent Vector Calculator?
A Unit Tangent Vector Calculator finds the direction of motion along a curve at a specific parameter value. Given a parametric curve r(t), it computes the velocity r′(t), the speed , and then normalizes to produce the unit tangent . The result is a length‑one vector pointing along the curve’s instantaneous direction. This is essential in multivariable calculus, physics (trajectory direction), computer graphics (path following), and differential geometry (Frenet–Serret frame). The calculator supports both 2D and 3D inputs and handles symbolic or numeric evaluation.
About the Unit Tangent Vector Calculator
The engine accepts components of r(t) such as , differentiates each, and evaluates at your chosen t. If the speed is zero (a cusp or stationary point), the unit tangent is undefined; the tool flags this and suggests nearby values. For arclength‑parameterized curves r(s), the tangent is simply . For sampled data, a centered finite difference approximates the derivative: , followed by normalization. Outputs include the raw derivative, speed, unit tangent, and optional rationalized surds at special angles.
Velocity:
Speed:
Unit tangent:
Arclength form:
Discrete derivative:
Undefined case:
How to Use This Unit Tangent Vector Calculator
- Enter the parametric components of r(t) (2D or 3D) and the parameter value t = t0.
- Choose symbolic or numeric mode. Optionally enable centered differences for sampled data.
- Submit to compute , , and . Copy components into your work.
- If speed is zero, adjust t or inspect the curve for cusps; consider limits from one side.
Examples
- 2D polynomial: . At : , , .
- Helix: ⇒ , , .
- Straight line: with constant ⇒ (constant).
- Sampled points: Given , compute , then normalize.
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FAQs
What does the unit tangent vector represent?
It’s the instantaneous direction along a curve—velocity direction with unit length—useful for motion, curvature, and path following.
Why is speed required?
Normalizing by speed ensures the vector has length one. Without it, you only have the unscaled velocity.
Can this handle 3D curves?
Yes. Enter x(t), y(t), and z(t). The same differentiation and normalization steps apply.
What if the derivative is zero?
The tangent is undefined at that parameter. Check nearby values or use one‑sided limits if the curve has a cusp.
Is the result independent of parameterization?
For increasing reparameterizations, the direction is preserved. Arclength parameterization makes automatically.
How precise are numeric results?
Precision depends on step size and rounding. Use exact mode for symbolic derivatives when possible.
Can I compute curvature next?
Yes. Curvature uses or after finding .
Does this work with piecewise curves?
Yes, compute the tangent on the active piece. At joints, evaluate limits from each side.
Can I input degrees for trigonometric curves?
Use radians for calculus by default. If you input degrees, convert first to avoid errors.
What about param ranges?
Ensure your chosen t lies in the domain of r(t). The calculator warns if components are undefined.
How do sampled‑data inputs work?
Provide t±h points. The tool uses a centered difference to estimate r′(t) and then normalizes.