Enter point and derivative data
This tool normalizes the derivative vector r′(t0). Use the point for tangent-line placement and the derivative components for direction.
Plotly graph
The graph shows the selected point and a centered tangent segment aligned with the normalized direction vector.
Tip: if z0 and dz/dt are both zero, the plot switches to a 2D view automatically.
Example data table
Use this worked example to verify your implementation, compare outputs, and understand how the tangent vector is normalized.
| Metric | Example value |
|---|---|
| Point r(t0) | (2, -1, 3) |
| Derivative r'(t0) | <3, 4, 12> |
| Speed ||r'(t0)|| | 13 |
| Unit tangent T(t0) | <0.230769, 0.307692, 0.923077> |
| Direction angles | α=76.686°, β=72.080°, γ=22.620° |
| Tangent line | x=2+0.230769s, y=-1+0.307692s, z=3+0.923077s |
Formula used
For a vector-valued curve r(t) = ⟨x(t), y(t), z(t)⟩, the unit tangent vector at t0 is found by normalizing the derivative:
The derivative magnitude is the speed or tangent-vector length:
The tangent line through the chosen point r(t0) uses the normalized direction:
Direction angles come from α = cos⁻¹(Tx), β = cos⁻¹(Ty), and γ = cos⁻¹(Tz). Because T(t0) is normalized, its length should equal 1.
How to use this calculator
- Enter the coordinates of the point where the tangent is required.
- Enter the derivative components dx/dt, dy/dt, and dz/dt at that same parameter value.
- Choose a plot half-length scale to control the tangent segment size.
- Select the number of decimal places for the displayed outputs.
- Press Compute Unit Tangent to show the result above the form.
- Review the normalized vector, direction angles, tangent line, and graph.
- Use the CSV or PDF buttons to export the worked result or example dataset.
Frequently asked questions
1. What does the unit tangent vector represent?
It shows the instantaneous direction of motion along a curve. Because it is normalized, its length is exactly one, so it describes direction without mixing in speed or scale.
2. Why must the derivative vector be nonzero?
Normalization divides by the derivative magnitude. If the derivative is the zero vector, the curve has no defined tangent direction at that point, so a unit tangent cannot be computed there.
3. Can this calculator handle 2D curves?
Yes. Set z0 and dz/dt to zero for planar problems. The calculator still uses the same normalization formula, then automatically draws a 2D Plotly view.
4. What is the difference between the tangent vector and unit tangent?
The tangent vector is r′(t0), which carries both direction and magnitude. The unit tangent divides that vector by its magnitude, keeping direction while forcing the vector length to equal one.
5. What do the direction angles mean?
They measure the angle between the unit tangent and the positive coordinate axes. Smaller angles indicate stronger alignment with that axis, while larger angles indicate weaker directional contribution.
6. Why is there a tangent line output?
The tangent line gives a local linear model of the curve near the selected point. It is useful for geometry, motion analysis, approximation, and visual verification of the tangent direction.
7. What does the normalization check confirm?
It confirms that the computed unit tangent has length one. Small display differences may appear from rounding, but the exact normalized vector should satisfy ||T|| = 1.
8. When should I change the plot scale?
Increase the scale when the tangent segment looks too short to inspect clearly. Reduce it when the segment dominates the view or hides the base point in a tight plotting range.