Tangent Vector Calculator

Calculator

Supports trigonometric, exponential, and power expressions. Use variables t, x, y as shown.

Accepted functions

sin cos tan asin acos atan sqrt abs exp log pow

Mode

Pick the curve representation.

Step size h

Central difference uses ±h around the point.

Direction option

Return unit tangent vector

When unchecked, returns raw derivative vector.

x(t)

y(t)

t₀

x(t)

y(t)

z(t)

t₀

f(x)

Use variable x.

x₀

F(x, y)

Use variables x and y.

x₀

y₀

Formula used

Parametric curve: If r(t)=<x(t), y(t)> then tangent vector is r′(t)=<dx/dt, dy/dt>. For 3D, include dz/dt.
Explicit curve: For y=f(x), direction <1, f′(x)> and slope is f′(x).
Implicit curve: For F(x,y)=0, the gradient ∇F=<F_x, F_y> is normal, so a tangent direction is <F_y, −F_x>.
Numerical derivative: Central difference f′(a) ≈ (f(a+h) − f(a−h)) / (2h).

How to use this calculator

Choose a mode that matches your curve: parametric, explicit, or implicit.
Enter the function expressions using the shown variable names.
Set the evaluation point (t₀ or x₀,y₀) and a small step size h.
Enable unit tangent if you want a normalized direction vector.
Press submit to see the tangent vector, magnitude, slope, and line form.

Example data table

Mode	Input	Point	Output tangent
Parametric (2D)	x=cos(t), y=sin(t), t₀=π/4	(0.7071, 0.7071)	<−0.7071, 0.7071>
Explicit	f(x)=x^2, x₀=2	(2, 4)	<1, 4>
Implicit	x^2+y^2−1=0, (0.6,0.8)	(0.6, 0.8)	<1.6, −1.2>
Parametric (3D)	x=cos(t), y=sin(t), z=t, t₀=1	(0.5403, 0.8415, 1)	<−0.8415, 0.5403, 1>

FAQs

1) What is a tangent vector?

A tangent vector is a direction that “just touches” a curve at a point. For parametric curves it is the derivative of the position vector, and for explicit curves it follows the derivative slope at that x-value.

2) Why do you use numerical differentiation?

It works for many formulas without symbolic algebra. Central differences are accurate for small step sizes, especially for smooth functions, and they avoid manual derivative mistakes while staying fast.

3) How do I choose a good step size h?

Start with 1e-4 or 1e-5 for typical inputs. If results look unstable, try a slightly larger h. If results look overly smoothed, try a smaller h, but too small can amplify rounding errors.

4) What does “unit tangent” mean?

A unit tangent vector has length 1. It keeps only the direction, not the speed of the parameter. This is useful for geometry, arc-length work, and for comparing directions across different parameterizations.

5) Why can the tangent be undefined?

If the derivative vector becomes zero, the curve has no unique tangent direction at that point. For implicit curves, a zero gradient means the normal is not defined, so tangent direction may not be unique.

6) Does the tangent vector depend on parameterization?

Yes. The raw derivative scales with how fast the parameter runs along the curve. However, the unit tangent direction usually matches the geometric direction, unless the parameterization reverses orientation.

7) What if my curve includes absolute values or roots?

You can use functions like abs() and sqrt(). Be careful at points where the function is not smooth, because derivatives may change abruptly and numeric estimates can vary with h.

8) How can I verify the output?

Check that the tangent line touches the curve near the point and that the direction makes sense. For implicit curves, confirm the tangent is perpendicular to the gradient by checking the dot product is near zero.