Quaternion Dot Product

Compute the dot product (scalar product) of two quaternions

Quaternion Dot Product Calculator

Quaternion Dot Product (Dot-Product)

Computes the dot product q₁ · q₂ of two quaternions by component-wise multiplication and summation into a real scalar value

Dot product properties

Formula: q₁ · q₂ = w₁w₂ + x₁x₂ + y₁y₂ + z₁z₂
Result: Real number (scalar), not a quaternion
Use: Angle between quaternions, measure similarity

Enter two quaternions for the dot product
First quaternion (q₁)
Second quaternion (q₂)
q₁ · q₂ = w₁w₂ + x₁x₂ + y₁y₂ + z₁z₂
Dot Product Result
Result (real scalar):
Dot product: q₁ · q₂ = w₁w₂ + x₁x₂ + y₁y₂ + z₁z₂ (real value)

Dot Product Info

Properties

Commutative: q₁ · q₂ = q₂ · q₁
Bilinear: (aq₁ + bq₂) · q₃ = a(q₁·q₃) + b(q₂·q₃)
Result: Real number (scalar)

Commutative Bilinear Scalar result

Angle: Relation to the angle between quaternions
Length: q · q = |q|² (norm squared)

Special cases
q · q = |q|²: Norm squared
q₁ · q₂ = 0: Orthogonal quaternions
q₁ · q₂ > 0: "Similar" orientation
q₁ · q₂ < 0: "Opposite" orientation

Formulas for the Quaternion Dot Product

Dot product formula
\[q_1 \cdot q_2 = w_1 w_2 + x_1 x_2 + y_1 y_2 + z_1 z_2\]

Sum of component-wise products

Vector representation
\[q_1 \cdot q_2 = \begin{pmatrix} w_1 \\ x_1 \\ y_1 \\ z_1 \end{pmatrix} \cdot \begin{pmatrix} w_2 \\ x_2 \\ y_2 \\ z_2 \end{pmatrix}\]

As a 4D vector dot product

Norm squared
\[q \cdot q = |q|^2 = w^2 + x^2 + y^2 + z^2\]

Dot product with itself

Angle between quaternions
\[\cos(\theta) = \frac{q_1 \cdot q_2}{|q_1| |q_2|}\]

For normalized quaternions

Algebraic properties
\[\begin{align} q_1 \cdot q_2 &= q_2 \cdot q_1 \\ (\lambda q_1) \cdot q_2 &= \lambda (q_1 \cdot q_2) \\ (q_1 + q_2) \cdot q_3 &= q_1 \cdot q_3 + q_2 \cdot q_3 \end{align}\]

Commutative, homogeneous, distributive

Examples for Quaternion Dot Product

Example 1: Basic computation
q₁ = 3 + 2i + 4j + 1k q₂ = 1 + 3i + 5j + 2k
Dot product calculation: \[\begin{align} q_1 \cdot q_2 &= w_1 w_2 + x_1 x_2 + y_1 y_2 + z_1 z_2 \\ &= 3 \cdot 1 + 2 \cdot 3 + 4 \cdot 5 + 1 \cdot 2 \\ &= 3 + 6 + 20 + 2 \\ &= 31 \end{align}\]

q₁ · q₂ = 31

Example 2: Norm squared
q = 2 + 3i + 1j + 4k
Norm squared: \[\begin{align} q \cdot q &= w^2 + x^2 + y^2 + z^2 \\ &= 2^2 + 3^2 + 1^2 + 4^2 \\ &= 4 + 9 + 1 + 16 \\ &= 30 \end{align}\] Norm: \[|q| = \sqrt{30} \approx 5.477\]

|q|² = 30

Example 3: Orthogonal quaternions
q₁ = 1 + 1i + 0j + 0k q₂ = 1 - 1i + 0j + 0k
Dot product: \[\begin{align} q_1 \cdot q_2 &= 1 \cdot 1 + 1 \cdot (-1) + 0 \cdot 0 + 0 \cdot 0 \\ &= 1 - 1 + 0 + 0 \\ &= 0 \end{align}\] Interpretation: Quaternions are orthogonal

Orthogonal: q₁ ⊥ q₂

Example 4: Angle between unit quaternions
q₁ = 1 + 0i + 0j + 0k (|q₁|=1) q₂ = 0.6 + 0.8i + 0j + 0k (|q₂|=1)
Dot product: \[q_1 \cdot q_2 = 1 \cdot 0.6 + 0 \cdot 0.8 = 0.6\] Angle: \[\cos(\theta) = 0.6 \Rightarrow \theta = \arccos(0.6) \approx 53.13°\]

Angle ≈ 53.13°

Geometric meaning
Similarity
Orientation comparison
Angle computation
Between rotations
Orthogonality
Perpendicularity
Projection
Directional component

The dot product measures "similarity" or the angle between two quaternions in 4D space

Step-by-step guide
Preparation
  1. Write both quaternions in standard form
  2. Identify all four components of each quaternion
  3. Pair components (w₁↔w₂, x₁↔x₂, etc.)
  4. Compute all four products individually
Computation
  1. W-terms: w₁ × w₂
  2. X-terms: x₁ × x₂
  3. Y-terms: y₁ × y₂
  4. Z-terms: z₁ × z₂, then sum all

Applications of the Quaternion Dot Product

The quaternion dot product is a versatile tool for various analyses:

3D Graphics & Animation
  • Angle computation: measure between rotations
  • Interpolation quality: determine SLERP parameter
  • Similarity comparison: compare orientations
  • Optimization: find shortest rotation paths
Robotics & Control
  • Orientation comparison: desired-actual deviation
  • Path planning: orientation continuity
  • Calibration: check sensor alignment
  • Stability: detect orientation drift
Mathematical analysis
  • Norm computation: |q|² = q·q
  • Orthogonality: perpendicular quaternions
  • Projection: components in directions
  • Normalization: check unit quaternions
Key properties
  • Result: Real number (scalar)
  • Commutativity: q₁·q₂ = q₂·q₁
  • Linearity: Distributive over addition
  • Geometry: Angle and similarity

Quaternion Dot Product: a window into 4D geometry

The quaternion dot product extends the familiar dot product concept from 3D vector algebra into the 4D space of quaternions. As a bilinear form it yields a real scalar that quantifies the "similarity" or the angle between two quaternions. In the context of 3D rotations the dot product enables measuring the angle between orientations, which is essential for interpolation algorithms, optimization and stability analysis. Its commutative and bilinear nature makes it a reliable tool for advanced quaternion computations.

Summary

The quaternion dot product combines the algebraic simplicity of component-wise multiplication with deep geometric insight. It enables quantitative analysis of quaternion relationships and is fundamental for algorithms such as SLERP where the angle between quaternions is crucial. Its role in computing norms, detecting orthogonality and measuring orientation similarity makes it indispensable in modern 3D mathematics. Understanding the dot product is key to mastering advanced quaternion techniques.

More Quaternion Functions

Addition  •  Subtraction  •  Division  •  Multiplication  •  Concatenate  •  Length  •  Interpolation  •  Normalize  •  Scalar Multiplication  •  Dot Product  •  Yaw-Pitch-Roll  •  Conjugates  •  Inverse  •  Negation  •