How to Calculate Magnitude of Vector

Vector Magnitude Calculator

Use this calculator to determine the magnitude (length) of a 2D or 3D vector. Enter the components of your vector below.

function calculateMagnitude() { var x = parseFloat(document.getElementById('xComponent').value); var y = parseFloat(document.getElementById('yComponent').value); var z = parseFloat(document.getElementById('zComponent').value); if (isNaN(x) || isNaN(y) || isNaN(z)) { document.getElementById('magnitudeResult').innerHTML = 'Please enter valid numbers for all components.'; return; } var magnitude = Math.sqrt(x * x + y * y + z * z); document.getElementById('magnitudeResult').innerHTML = '

Magnitude: ' + magnitude.toFixed(4) + '

'; }

Understanding Vector Magnitude

In mathematics and physics, a vector is a quantity that has both magnitude and direction. Unlike scalar quantities (like temperature or mass) which only have magnitude, vectors describe movement, force, velocity, and other directional phenomena. The magnitude of a vector is essentially its length or size, representing the "strength" or "amount" of the quantity it describes, irrespective of its direction.

What is a Vector?

A vector can be visualized as an arrow pointing from one point to another. It's often represented by its components along coordinate axes. For example, a 2D vector might be written as v = (x, y) or xi + yj, where x and y are its components along the X and Y axes, respectively. A 3D vector would be v = (x, y, z) or xi + yj + zk.

The Formula for Magnitude

The magnitude of a vector is calculated using the Pythagorean theorem. If a vector v has components (x, y, z), its magnitude, denoted as ||v|| or |v|, is given by the formula:

||v|| = √(x² + y² + z²)

For a 2D vector (x, y), the formula simplifies to:

||v|| = √(x² + y²) (where z = 0)

How to Use This Calculator

1. Enter X-Component (i): Input the value of the vector's component along the X-axis.
2. Enter Y-Component (j): Input the value of the vector's component along the Y-axis.
3. Enter Z-Component (k): Input the value of the vector's component along the Z-axis. If you are calculating the magnitude of a 2D vector, simply enter '0' for the Z-Component.
4. Click "Calculate Magnitude": The calculator will instantly display the magnitude of your vector.

Examples:

• Example 1: 2D Vector
  Vector v = (3, 4)
  X-Component: 3
  Y-Component: 4
  Z-Component: 0
  Calculation: √(3² + 4² + 0²) = √(9 + 16) = √25 = 5
  Magnitude: 5
• Example 2: 3D Vector
  Vector u = (1, -2, 2)
  X-Component: 1
  Y-Component: -2
  Z-Component: 2
  Calculation: √(1² + (-2)² + 2²) = √(1 + 4 + 4) = √9 = 3
  Magnitude: 3
• Example 3: Vector with Negative Components
  Vector w = (-5, -12, 0)
  X-Component: -5
  Y-Component: -12
  Z-Component: 0
  Calculation: √((-5)² + (-12)² + 0²) = √(25 + 144) = √169 = 13
  Magnitude: 13

Understanding vector magnitude is fundamental in many scientific and engineering fields, from calculating the speed of an object (velocity magnitude) to determining the strength of a force.