Solve tangent and normal vectors from motion data. Use 2D or 3D inputs with ease. See curvature, components, exports, and examples in one place.
| Case | Dimension | Velocity v | Acceleration a | Unit Tangent T | Unit Normal N |
|---|---|---|---|---|---|
| Example 1 | 2D | (3, 4) | (-4, 3) | (0.6000, 0.8000) | (-0.8000, 0.6000) |
| Example 2 | 3D | (2, -1, 2) | (1, 2, -1) | (0.6667, -0.3333, 0.6667) | (0.5963, 0.7454, -0.2981) |
The unit tangent vector comes from the velocity vector. First compute speed as |v| = √(vx² + vy² + vz²). Then divide each velocity component by the speed.
T = v / |v|
The tangential acceleration scalar is the dot product of acceleration with the unit tangent vector.
aT = a · T
The normal acceleration vector is the remaining part of acceleration after removing the tangential part.
anormal = a − (a · T)T
The unit normal vector is the normalized normal acceleration vector.
N = anormal / |anormal|
The curvature is computed with:
κ = aN / |v|²
Choose whether your problem is in 2D or 3D.
Enter the velocity vector components at the selected parameter value.
Enter the acceleration vector components at the same point.
Select the number of decimal places you want.
Press the calculate button.
Read the result table placed above the form. It shows the unit tangent vector, unit normal vector, tangential acceleration, normal acceleration, curvature, and turning rate.
Use the CSV button for spreadsheet-ready output. Use the PDF button for a printable summary.
Unit tangent and unit normal vectors describe motion along a curve. They turn raw derivatives into geometric meaning. The tangent vector shows the direction of travel. The normal vector shows where the path is bending next.
This calculator uses velocity and acceleration data at one point. It first finds the speed. Then it normalizes the velocity vector to build the unit tangent vector. After that, it removes the tangential part of acceleration. The remaining part becomes the normal acceleration vector. Once normalized, that vector gives the unit normal direction.
Curvature tells you how sharply a path changes direction. A larger curvature means a tighter bend. A small curvature means the path is nearly straight. This is useful in calculus, mechanics, robotics, and trajectory design. It also helps in engineering models that track turning behavior.
Sometimes the calculator reports that the unit normal vector is undefined. This happens when the normal component of acceleration is zero. In that case, the path is locally straight at that instant, or the available data does not show directional bending. The tangent vector can still exist if the speed is nonzero.
Students often learn the formulas but miss the interpretation. This tool connects both parts. You can compare velocity, tangential acceleration, normal acceleration, and curvature in one place. That makes homework checking faster and clearer.
Use this calculator for parametric curves, motion analysis, and vector calculus exercises. It works well for two-dimensional and three-dimensional cases. The export tools also help when you need a record for class notes, reports, or solved examples.
The unit tangent vector is the normalized velocity vector. It points in the direction of motion and always has magnitude one.
The unit normal vector points toward the direction of turning. It is perpendicular to the unit tangent vector when the curve bends normally.
If the normal part of acceleration is zero, there is no bending direction to normalize. The motion is locally straight at that instant.
Yes. Select 2D for planar motion or 3D for space curves. The calculator adjusts the vector inputs automatically.
If the velocity magnitude is zero, the unit tangent vector cannot be formed. The calculator stops because direction from zero speed is undefined.
Tangential acceleration measures how fast the speed changes along the path. It is the projection of acceleration onto the unit tangent vector.
Normal acceleration measures how fast the direction changes. It controls the bending behavior of the path rather than the speed itself.
Curvature summarizes how sharply a curve turns. It helps connect derivatives, geometry, and motion in one interpretable value.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.