Quaternion Scalar Multiplication

Multiply a quaternion by a real scalar

Quaternion Scalar Multiplication Calculator

Quaternion Scalar Multiplication

Multiplies a quaternion q with a real scalar λ by component-wise multiplication of all four components

Scalar multiplication

Simple operation: λ × q = λ × (w + xi + yj + zk)
Commutative: λ × q = q × λ (scalar may be left or right)
Scaling: changes the "size" of the quaternion without altering direction

Enter quaternion and scalar
Quaternion (q)
Multiplier
Real multiplication factor
Operation overview
λ × q = λ × (w + xi + yj + zk) = λw + λxi + λyj + λzk
Each component is multiplied by the scalar
Scalar multiplication result
W (scalar):
X (i comp.):
Y (j comp.):
Z (k comp.):
Scalar multiplication: λ × q = (λw, λx, λy, λz)

Scalar multiplication info

Properties

Commutative: λ × q = q × λ
Associative: λ × (μ × q) = (λμ) × q
Distributive: λ × (q₁ + q₂) = λq₁ + λq₂

Commutative Associative Simple

Simple: Each component × scalar
Scaling: changes "size", not direction

Special cases
λ = 1: quaternion unchanged
λ = 0: zero quaternion (0,0,0,0)
λ = -1: negated quaternion
λ = 1/|q|: normalization


Formulas for quaternion scalar multiplication

General scalar multiplication formula
\[\lambda \times q = \lambda \times (w + xi + yj + zk) = \lambda w + \lambda xi + \lambda yj + \lambda zk\]

Component-wise multiplication with the scalar

Component-wise representation
\[\begin{align} w' &= \lambda \cdot w \\ x' &= \lambda \cdot x \\ y' &= \lambda \cdot y \\ z' &= \lambda \cdot z \end{align}\]

Each component multiplied individually

Vector representation
\[\lambda \times \begin{pmatrix} w \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \lambda w \\ \lambda x \\ \lambda y \\ \lambda z \end{pmatrix}\]

As a 4D vector scalar multiplication

Magnitude of scaled quaternion
\[|\lambda \times q| = |\lambda| \times |q|\]

Magnitude scales with the absolute scalar value

Algebraic properties
\[\begin{align} \lambda \times q &= q \times \lambda \\ (\lambda \mu) \times q &= \lambda \times (\mu \times q) \\ \lambda \times (q_1 + q_2) &= \lambda q_1 + \lambda q_2 \end{align}\]

Commutative, associative, distributive

Examples for quaternion scalar multiplication

Example 1: Simple scaling
q = 1 + 3i + 5j + 2k λ = 2
Scalar multiplication: \[\begin{align} \lambda \times q &= 2 \times (1 + 3i + 5j + 2k) \\ w' &= 2 \times 1 = 2 \\ x' &= 2 \times 3 = 6 \\ y' &= 2 \times 5 = 10 \\ z' &= 2 \times 2 = 4 \end{align}\]

λ × q = 2 + 6i + 10j + 4k

Example 2: Negative scalar
q = 2 + 4i - 3j + 1k λ = -0.5
Calculation: \[\begin{align} w' &= -0.5 \times 2 = -1 \\ x' &= -0.5 \times 4 = -2 \\ y' &= -0.5 \times (-3) = 1.5 \\ z' &= -0.5 \times 1 = -0.5 \end{align}\]

λ × q = -1 - 2i + 1.5j - 0.5k

Example 3: Normalization
q = 3 + 4i + 0j + 0k |q| = 5
Normalization scalar: \[\lambda = \frac{1}{|q|} = \frac{1}{5} = 0.2\] Normalized quaternion: \[\begin{align} \hat{q} &= 0.2 \times (3 + 4i) \\ &= 0.6 + 0.8i \end{align}\] Verification: \[|\hat{q}| = \sqrt{0.6^2 + 0.8^2} = 1\]

Unit quaternion generated

Example 4: Special cases
q = 1 + 2i + 3j + 4k
λ = 0: \[0 \times q = 0 + 0i + 0j + 0k\] λ = 1: \[1 \times q = q \text{ (unchanged)}\] λ = -1: \[-1 \times q = -1 - 2i - 3j - 4k\]

Zero, identity, negation

Geometric meaning
Scaling
Size change
Normalization
Unit length
Sign flip
λ = -1
Interpolation
Weighting

Scalar multiplication changes the "size" of the quaternion but not its "direction" in 4D space

Step-by-step guide
Preparation
  1. Write quaternion in standard form
  2. Choose scalar value λ
  3. Identify all four components
  4. Compute systematically
Execution
  1. W component: λ × w
  2. X component: λ × x
  3. Y component: λ × y
  4. Z component: λ × z

Anwendungen der Quaternion-Skalarmultiplikation

Die Quaternion-Skalarmultiplikation ist eine grundlegende Operation mit vielen Anwendungen:

3D-Grafik & Animation
  • Normalisierung: Einheits-Quaternionen erzeugen
  • Skalierung: Rotations-"Intensität" ändern
  • Interpolation: Gewichtete Mischung
  • Animation: Zeitbasierte Skalierung
Robotik & Steuerung
  • Kalibrierung: Sensordaten normalisieren
  • Dämpfung: Rotationsgeschwindigkeit reduzieren
  • Verstärkung: Signal-Amplifikation
  • Filterung: Gewichtete Mittelwerte
Numerical mathematics
  • Normalization: enforce |q| = 1
  • Scaling: adjust magnitude
  • Sign flip: λ = -1
  • Discretization: scale time steps
Key properties
  • Simplicity: only multiplication required
  • Commutativity: λ×q = q×λ
  • Linearity: distributive over addition
  • Magnitude: |λ×q| = |λ|×|q|

Quaternion scalar multiplication: simple but powerful

The quaternion scalar multiplication is the simplest operation in quaternion algebra, yet one of the most useful. It enables proportional scaling of all quaternion components by a real factor. This operation is commutative, associative and distributive, making it ideal for normalization, weighting and linear transformations. In 3D graphics it is frequently used to generate unit quaternions required for valid rotations. The operation preserves the "direction" of the quaternion in 4D space and only changes its "size".

Summary

Quaternion scalar multiplication combines simplicity with versatility. As a component-wise multiplication it is computationally trivial, but its applications range from basic normalization to complex interpolation algorithms. Its algebraic properties (commutativity, associativity, distributivity) make it a reliable building block for larger quaternion operations. In modern 3D applications scalar multiplication is indispensable for generating and manipulating unit quaternions that form the backbone of rotation calculations.




More Quaternion Functions

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