About

Clamping a vector restricts its magnitude ‖v‖ to a defined interval [min, max] while preserving its direction. This operation appears constantly in game physics (limiting velocity), robotics (constraining actuator forces), shader programming (bounding color intensities), and AI gradient descent (gradient clipping prevents exploding gradients). Getting it wrong causes subtle bugs: an unclamped velocity vector lets objects tunnel through walls; an unclamped force vector damages servo motors. The zero-vector edge case is critical. When ‖v‖ = 0 and min > 0, no unit direction exists, so the result must remain the zero vector. This tool computes exact clamped components for 2D and 3D vectors and renders both vectors on a live canvas.

The tool assumes Euclidean (L2) norm. It does not apply to L1 or L∞ norms. Component-wise clamping is a different operation entirely. If you need per-axis clamping, apply clamp to each component independently. Pro tip: in real-time simulations, clamp before integration to avoid a single oversized timestep introducing instability.

Formulas

The magnitude (Euclidean norm) of a vector v = (x, y, z) is computed as:

‖v‖ = √x² + y² + z²

The clamped magnitude is restricted to the interval:

m = clamp(‖v‖, min, max) =

{

min if ‖v‖ < minmax if ‖v‖ > max‖v‖ otherwise

The clamped vector preserves direction by scaling:

v_clamped = v ⋅ m‖v‖

When ‖v‖ = 0, division is undefined. The result is the zero vector 0 regardless of bounds.

Where: x, y, z = vector components; min = minimum allowed magnitude; max = maximum allowed magnitude; m = clamped magnitude; ‖v‖ = original Euclidean magnitude.

Reference Data

Domain	Typical Vector	Common Clamp Range	Why Clamp
Game Physics - Velocity	Player velocity m/s	[0, 50] m/s	Prevent tunneling through colliders
Game Physics - Force	Applied impulse N	[0, 1000] N	Stability of physics solver
Robotics - Torque	Joint torque N⋅m	[0.1, 5.0] N⋅m	Protect servo motors from burnout
ML - Gradient Clipping	Gradient vector	[0, 1.0]	Prevent exploding gradients
Shaders - Color	RGB intensity	[0, 1]	HDR → LDR tone mapping
Audio DSP	Sample amplitude vector	[0, 0.95]	Prevent digital clipping distortion
Navigation - GPS	Velocity vector km/h	[0, 130] km/h	Filter GPS noise spikes
Camera Control	Look direction	[0.01, 1]	Prevent degenerate zero-length look vector
Cloth Simulation	Spring displacement	[0, 0.5] m	Prevent mesh explosion
Drone Control	Thrust vector N	[2, 20] N	Stay within motor thrust envelope
Particle Systems	Emission velocity	[1, 10] m/s	Visual consistency
Electromagnetic Sim	E-field vector V/m	[0, 3×10⁶]	Dielectric breakdown threshold
Wind Simulation	Wind velocity m/s	[0, 75]	Category 5 hurricane cap
Cursor/Pointer	Mouse delta px	[0, 100] px/frame	Smooth input, reject jitter
Fluid Dynamics (SPH)	Particle velocity	[0, 340] m/s	CFL condition compliance

Frequently Asked Questions

A zero vector has no direction. You cannot scale a zero-length vector to any non-zero magnitude because the unit vector is undefined (division by zero). The tool returns the zero vector and flags this edge case. In practice, you should handle this upstream by assigning a default direction before clamping.

Vector clamping restricts the overall magnitude ‖v‖ while preserving direction. Component-wise clamping applies clamp independently to each axis (x, y, z), which changes the direction. For a vector (3, 4) clamped component-wise to [0, 2], you get (2, 2) - a completely different angle. Magnitude clamping yields (1.2, 1.6) if max = 2, preserving the original 53.13° heading.

No. The tool validates that min ≤ max. If min exceeds max, the clamp interval is degenerate and mathematically ambiguous. The tool shows an error and does not compute. Fix the bounds before proceeding.

No. This tool uses the Euclidean (L2) norm exclusively: ‖v‖₂ = √∑x_i². L1 norm (Manhattan) and L∞ norm (Chebyshev) require different scaling logic. For L∞, you would clamp based on the maximum absolute component, not the quadratic sum.

The clamping formula multiplies the original vector by the scalar ratio m‖v‖. Since this is a positive scalar multiplication, it scales all components uniformly, preserving the angle. The unit direction vector v‖v‖ remains identical.

During backpropagation, gradient vectors can grow exponentially (exploding gradients). Gradient clipping applies magnitude clamping with max typically set to 1.0 or 5.0. If ‖g‖ > max, the gradient is rescaled to g ⋅ max‖g‖. This preserves the descent direction while bounding the step size.