Vector Scalar Multiplication

Calculator and formulas for multiplying a vector by a scalar (real number)

Vector Scalar Multiplication Calculator

Vector by Scalar Multiplication

Multiplies a vector v by a scalar k: k × v = [k×x, k×y, k×z]

Select Vector Dimension
Vector to Multiply (v)
X value of vector
Y value of vector
Z value of vector
W value of vector
Multiplier (Scalar k)
Real number (multiplier)
k > 0: Direction preserved,     k < 0: Direction reversed
Scalar Multiplication Result
X:
Y:
Z:
W:
Scaled Vector:
Each component is multiplied by the scalar: k × v = [k×x, k×y, ...]

Scalar Multiplication Info

Scalar Multiplication Properties

Scaling: Changes the length of the vector

k > 0: Direction ↑ k < 0: Direction ↓ Scalar

Enlargement: |k| > 1 makes vector longer
Reduction: |k| < 1 makes vector shorter

Examples
[2, 3, 4] × 5 = [10, 15, 20]
[1, -2] × (-3) = [-3, 6]
[x, y, z] × 0 = [0, 0, 0]

Formulas for Vector-Scalar Multiplication

Basic Formula
\[k \cdot \vec{v} = k \cdot \left[\matrix{x\\y\\z}\right]\]

Scalar multiplied by vector

2D Scalar Multiplication
\[k \cdot \left[\matrix{x\\y}\right] = \left[\matrix{k \cdot x\\k \cdot y}\right]\]

Component-wise multiplication

3D Scalar Multiplication
\[k \cdot \left[\matrix{x\\y\\z}\right] = \left[\matrix{k \cdot x\\k \cdot y\\k \cdot z}\right]\]

Three-dimensional scaling

Magnitude After Multiplication
\[|k \cdot \vec{v}| = |k| \cdot |\vec{v}|\]

Magnitude is multiplied by |k|

Calculation Examples for Vector-Scalar Multiplication

Example 1: Positive Scaling
[2, 3, 4] × 5
\[5 \cdot \left[\matrix{2\\3\\4}\right] = \left[\matrix{5 \cdot 2\\5 \cdot 3\\5 \cdot 4}\right] = \left[\matrix{10\\15\\20}\right]\]

Result: [10, 15, 20] - Vector becomes 5× longer

Example 2: Negative Scaling
[3, -6] × (-2)
\[-2 \cdot \left[\matrix{3\\-6}\right] = \left[\matrix{-2 \cdot 3\\-2 \cdot (-6)}\right] = \left[\matrix{-6\\12}\right]\]

Result: [-6, 12] - Direction reversed, 2× longer

Geometric Interpretation
k > 1: Enlargement
Vector becomes longer
0 < k < 1: Reduction
Vector becomes shorter
k < 0: Reversal
Direction is reversed
k = 0: Zero Vector
Vector becomes [0,0,...]

Multiplication scales the vector proportionally and can reverse its direction

Special Cases and Important Values
k = 1

• v × 1 = v

• Vector remains unchanged

• Identity operation

k = 0

• v × 0 = [0, 0, ...]

• Becomes zero vector

• Length becomes zero

k = -1

• v × (-1) = -v

• Direction is reversed

• Same length

k = 0.5

• v × 0.5 = v/2

• Halves the length

• Direction preserved

Applications of Vector-Scalar Multiplication

Vector-scalar multiplication is a fundamental operation in many fields:

Computer Graphics & Animation
  • 3D object scaling and resizing
  • Adjust velocity vectors
  • Camera zoom and perspective
  • Scale lighting intensity
Physics & Engineering
  • Amplify or reduce force vectors
  • Velocities and accelerations
  • Electric and magnetic fields
  • Moment and torque calculations
Mathematics & Data Processing
  • Linear algebra and transformations
  • Feature scaling in machine learning
  • Statistical weightings
  • Signal processing and amplification
Robotics & Automation
  • Control movement velocities
  • Force and torque regulation
  • Path planning and trajectories
  • Calibrate sensor data

Vector-Scalar Multiplication: Proportional Scaling in Vector Space

Vector-scalar multiplication is a fundamental operation of linear algebra that multiplies a vector by a real number (scalar). This operation effects a uniform scaling of all vector components and thus changes the length of the vector proportionally to the magnitude of the scalar. For positive scalars, the direction is preserved; for negative ones, it is reversed. This elegant property makes the operation indispensable for scaling, amplification, and direction reversal in diverse application areas.

Summary

Vector-scalar multiplication combines mathematical elegance with practical versatility. The intuitive rule - multiply each component by the scalar - enables precise size and direction adjustments in arbitrary dimensions. From 3D graphics through physical simulations to robot control, scalar multiplication provides a direct method for proportional vector manipulation. It demonstrates how fundamental mathematical operations elegantly and efficiently solve complex scaling, amplification, and control problems.




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