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.

Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?




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