How to Calculate Magnitude

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Vector Magnitude Calculator

Result:

Magnitude (|v|): 0

What is Magnitude?

In mathematics and physics, magnitude refers to the size or length of a vector, regardless of its direction. It is a scalar quantity, meaning it is represented by a single number. Whether you are dealing with displacement, velocity, or force, calculating the magnitude is a fundamental step in vector analysis.

The Magnitude Formula

The magnitude of a vector is calculated using the Pythagorean Theorem. For a vector v with components (x, y, z), the formula for magnitude (denoted as |v|) is:

|v| = √(x² + y² + z²)

Step-by-Step Calculation Example

Let's say you have a 3D vector with the following components: X = 3, Y = 4, and Z = 5. To find the magnitude:

  1. Square each component: 3² = 9, 4² = 16, 5² = 25.
  2. Sum the squares: 9 + 16 + 25 = 50.
  3. Take the square root: √50 ≈ 7.071.

The magnitude of the vector (3, 4, 5) is approximately 7.071.

Magnitude in Different Fields

While the calculator above focuses on Euclidean vectors, the term "magnitude" is used in various scientific contexts:

  • Physics: Used to calculate the total force applied to an object or the speed of a moving body.
  • Seismology: The Richter scale measures the magnitude of earthquakes based on the amplitude of waves recorded by seismographs.
  • Astronomy: Refers to the brightness of stars. Apparent magnitude measures how bright a star looks from Earth, while absolute magnitude measures its intrinsic luminosity.
function calculateMagnitude() { var x = parseFloat(document.getElementById('compX').value) || 0; var y = parseFloat(document.getElementById('compY').value) || 0; var z = parseFloat(document.getElementById('compZ').value) || 0; var sumOfSquares = (x * x) + (y * y) + (z * z); var magnitude = Math.sqrt(sumOfSquares); var resultArea = document.getElementById('magnitudeResultArea'); var output = document.getElementById('magnitudeOutput'); var steps = document.getElementById('formulaSteps'); if (isNaN(magnitude)) { output.innerText = "Error"; steps.innerText = "Please enter valid numeric values."; } else { output.innerText = magnitude.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); steps.innerText = "Calculation: √(" + x + "² + " + y + "² + " + z + "²) = √(" + sumOfSquares + ")"; } resultArea.style.display = 'block'; }

Understanding how to calculate magnitude is essential for students, engineers, and data scientists. By breaking a vector down into its individual components, you can use the distance formula to find its total length. This tool simplifies that process, providing instant results for 2D or 3D vector problems.

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