Vector Reflection Calculator

Calculator and formulas for reflecting a vector across a surface with given normal

Vector Reflection Calculator

Vector Reflection (Refraction)

Calculates the reflection of vector v across a surface with normal vector n: r = v - 2(v·n)n

Reflection Across a Surface

The first vector is reflected across the surface whose normal vector is given by the second vector. The result is the reflected vector.

Select Vector Dimension
Reflection in the plane
Reflection in 3D space
Vector to Reflect (v)
Normal Vector (n)
Note: The normal vector does not need to be normalized
Reflection Result
X:
Y:
Z:
Reflected Vector:
Reflection formula: r = v - 2(v·n̂)n̂ (with normalized normal vector n̂)

Reflection Info

Reflection Properties

Reflection: Vector is reflected across a surface

Reflection Normal Vector Surface

Preservation: Magnitude remains same |r| = |v|
Angle: Angle of incidence = angle of reflection

Laws of Reflection
📐 Angle of incidence = Angle of reflection
📏 Same magnitude: |v| = |r|
⊥ Normal perpendicular to surface
Application Examples
Optics: Light reflection in mirrors
Billiards: Ball reflection on cushions
3D Graphics: Ray tracing

Formulas for Vector Reflection

Reflection Formula
\[\vec{r} = \vec{v} - 2(\vec{v} \cdot \hat{n})\hat{n}\]

With normalized normal vector n̂

General Form
\[\vec{r} = \vec{v} - 2\frac{\vec{v} \cdot \vec{n}}{|\vec{n}|^2}\vec{n}\]

For arbitrary normal vector n

Dot Product
\[\vec{v} \cdot \vec{n} = v_x n_x + v_y n_y + v_z n_z\]

For computing the projection

Magnitude Preservation
\[|\vec{r}| = |\vec{v}|\]

Magnitude remains constant during reflection

Calculation Examples for Vector Reflection

Example 1: 2D Reflection
v=[3,4], n=[1,0]
\[\begin{aligned} \vec{v} \cdot \hat{n} &= 3 \cdot 1 + 4 \cdot 0 = 3 \\ \vec{r} &= [3,4] - 2 \cdot 3 \cdot [1,0] \\ &= [3,4] - [6,0] = [-3,4] \end{aligned}\]

Reflection across the y-axis

Example 2: 3D Reflection
v=[1,-4,5], n=[0,1,0]
\[\begin{aligned} \vec{v} \cdot \hat{n} &= 1 \cdot 0 + (-4) \cdot 1 + 5 \cdot 0 = -4 \\ \vec{r} &= [1,-4,5] - 2 \cdot (-4) \cdot [0,1,0] \\ &= [1,-4,5] + [0,8,0] = [1,4,5] \end{aligned}\]

Reflection across the xz-plane

Geometric Interpretation
n⊥Surface
Normal perpendicular
α_in = α_out
Equal angles
|v| = |r|
Magnitude preservation
r·n ≤ 0
Direction reversal

Reflection follows classical laws of reflection

Special Cases of Reflection
v ⊥ n

• v·n = 0

• No reflection

• r = v (parallel to surface)

v ∥ n

• Perpendicular incidence

• r = -v

• Complete reversal

45° Angle

• Diagonal reflection

• Classic case

• Like billiard ball

n not normalized

• Adjust formula

• Divide by |n|²

• Automatically correct

Applications of Vector Reflection

Vector reflection is fundamental in many areas of physics, optics and computer graphics:

Optics & Lighting
  • Light reflection on mirrors and surfaces
  • Laws of reflection in optics
  • Laser and ray optics
  • Camera lenses and telescopes
Computer Graphics
  • Ray tracing and rendering
  • Specular reflection (gloss)
  • 3D mirror effects and reflections
  • Realistic surface shading
Physics & Mechanics
  • Ball sports: reflection of balls
  • Collision detection and response
  • Wave reflection (sound, water)
  • Elastic collisions and bouncing
Engineering
  • Radar and sonar systems
  • Antenna technology and signal reflection
  • Acoustics: sound reflection
  • Architecture: light guidance and reflection

Vector Reflection: Reflection in Linear Algebra

Vector reflection is a fundamental geometric transformation that reflects a vector across a plane or line. This operation follows the classical laws of reflection from physics: the angle of incidence equals the angle of reflection, the magnitude remains preserved, and the reflection occurs with respect to the normal to the reflecting surface. The mathematical elegance of the formula r = v - 2(v·n̂)n̂ combines vector algebra with geometric intuition and finds broad application from optics to computer graphics.

Summary

Vector reflection combines mathematical precision with physical intuition. The compact reflection formula encodes the fundamental laws of reflection and enables precise calculations in arbitrary dimensions. From simulating light reflection through collision detection to realistic 3D rendering, vector reflection forms the mathematical foundation for numerous technical applications. It demonstrates how elegant algebraic formulas can exactly describe complex physical phenomena and make them computationally tractable.




Vector Functions

Addition • Subtraction • Multiplication • Scalar Multiplication • Division • Scalar Division • Dot Product • Cross Product • Interpolation • Distance • Distance Squaret • Normalization • Reflection • Magnitude • Squared-Magnitude • Triple-Product