Result
Step-by-step solution
Vector details
Visualization
The plot uses automatic scaling and a rotatable Cartesian projection. Higher-dimensional vectors display their first three components.
Calculator inputs
Calculation history
Formula used
Vector magnitude and normalization
||v|| = √(v₁² + v₂² + ... + vₙ²) û = v / ||v||
Two-dimensional normals
n₁ = (-y, x) / √(x² + y²) n₂ = (y, -x) / √(x² + y²)
Three-dimensional plane normal
n = A × B n̂ = n / ||n||
Implicit surface normal
∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z) n̂ = ∇F / ||∇F||
How to use this calculator
- Choose the calculation mode that matches your vector, line, plane, surface, curve, or batch problem.
- Enter every required component or equation, then choose precision, angle units, and normal orientation.
- Select Calculate to display the exact setup, decimal result, verification checks, and vector visualization.
- Use the export controls to copy results, download CSV data, print a PDF, or create a shareable link.
- Review saved calculations in the local history section. History remains in the current browser until cleared.
Example data
| Mode | Input | Main operation | Expected result |
|---|---|---|---|
| Normalize | (3, 4) | Divide by magnitude 5 | (0.6, 0.8) |
| 2D normal | (3, 4) | Rotate and normalize | (-0.8, 0.6) or (0.8, -0.6) |
| Cross product | (1,0,0), (0,1,0) | A × B | (0,0,1) |
| Plane | 2x - y + 2z = 5 | Normalize (2,-1,2) | (2/3,-1/3,2/3) |
| Surface | x²+y²+z²-25 at (3,4,0) | Normalize gradient | (0.6,0.8,0) |
Understanding normal and unit vectors
A unit vector has magnitude one and preserves direction. Normalization divides every component by the original magnitude. The resulting vector is useful wherever direction must remain independent of scale.
A normal vector is perpendicular to another geometric object. In two dimensions, rotating a vector by ninety degrees creates a normal direction. In three dimensions, a cross product creates a vector perpendicular to two nonparallel vectors.
Planes have normal vectors built directly from their coefficients. For a plane written as Ax plus By plus Cz equals D, the vector (A, B, C) is normal. Dividing that vector by its magnitude gives a unit normal.
Implicit surfaces use the gradient as their local normal. The gradient points toward the fastest increase in the function value. This calculator estimates partial derivatives numerically when a surface equation is entered.
Parametric curves require both tangent and curvature information. The derivative gives the tangent direction, while the perpendicular acceleration component gives the principal normal. A zero curvature warning appears when that normal is undefined.
Verification helps detect input mistakes and rounding problems. A normalized vector should have magnitude one within the selected tolerance. Perpendicular vectors should have a dot product close to zero.
Orientation matters in graphics, engineering, and geometry. Reversing a normal changes its direction without changing perpendicularity. Right-hand and left-hand selections help match coordinate conventions and surface-facing requirements.
Projection separates one vector into parallel and perpendicular parts. The projection follows a reference vector, while rejection measures the remaining perpendicular component. Together, both parts reconstruct the original vector exactly before rounding.
Coordinate conversions make the calculator useful for polar, cylindrical, and spherical inputs. Angles may be entered in degrees or radians. Converted Cartesian components are shown before the normalization steps begin.
Frequently asked questions
What is a normal unit vector?
It is a perpendicular vector whose magnitude equals one.
Can a zero vector be normalized?
No. Its magnitude is zero, so division is undefined.
Why are there two normal directions?
A vector and its negative are both perpendicular and normalized.
How does the calculator detect perpendicular vectors?
It compares their dot product with the selected numerical tolerance.
What happens when two 3D vectors are parallel?
Their cross product is zero, so no unique unit normal exists.
Can I calculate a surface normal?
Yes. Enter an implicit function and the point where its gradient is evaluated.
Are spherical coordinates supported?
Yes. Enter radius, azimuth, and inclination using your selected angle unit.
How accurate are numerical derivatives?
Accuracy depends on the function, point, derivative step, and floating-point rounding.
Can I export several vector results?
Yes. Batch mode accepts pasted rows or uploaded CSV data and exports calculated rows.