Vector Calculator

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Vector Calculator (2D)

Vector A

Vector B

Enter values and select an operation to see results.
function calculateVector(op) { var ax = parseFloat(document.getElementById('vax').value) || 0; var ay = parseFloat(document.getElementById('vay').value) || 0; var bx = parseFloat(document.getElementById('vbx').value) || 0; var by = parseFloat(document.getElementById('vby').value) || 0; var resultDiv = document.getElementById('vector-result'); var output = ""; if (op === 'add') { var rx = ax + bx; var ry = ay + by; output = "Result (A + B): Vector R = (" + rx.toFixed(2) + ", " + ry.toFixed(2) + ")"; } else if (op === 'sub') { var rx = ax – bx; var ry = ay – by; output = "Result (A – B): Vector R = (" + rx.toFixed(2) + ", " + ry.toFixed(2) + ")"; } else if (op === 'dot') { var dot = (ax * bx) + (ay * by); output = "Dot Product (A · B): Scalar Value = " + dot.toFixed(2); } else if (op === 'mag') { var magA = Math.sqrt(ax * ax + ay * ay); var magB = Math.sqrt(bx * bx + by * by); output = "Magnitudes: |A| = " + magA.toFixed(4) + " |B| = " + magB.toFixed(4); } resultDiv.innerHTML = '
' + output + '
'; } function resetVectorCalc() { document.getElementById('vax').value = 0; document.getElementById('vay').value = 0; document.getElementById('vbx').value = 0; document.getElementById('vby').value = 0; document.getElementById('vector-result').innerHTML = 'Enter values and select an operation to see results.'; }

Understanding Vector Calculations

In physics and mathematics, a vector is an object that has both magnitude and direction. This vector calculator allows you to perform essential 2D vector operations, including addition, subtraction, dot product, and magnitude calculations, which are fundamental in fields like engineering, navigation, and computer graphics.

Common Vector Operations

1. Vector Addition (A + B): This operation combines two vectors into a resultant vector. Geometrically, it follows the "tip-to-tail" rule. Algebraically, you simply add the corresponding components:

  • Rx = Ax + Bx
  • Ry = Ay + By

2. Vector Subtraction (A – B): Subtraction is the addition of a negative vector. It represents the vector pointing from the tip of B to the tip of A:

  • Rx = Ax – Bx
  • Ry = Ay – By

3. Dot Product (A · B): The dot product is a scalar value (a simple number, not a vector). It is useful for finding the angle between vectors or determining if they are perpendicular (orthogonal). If the dot product is zero, the vectors are perpendicular.

  • Scalar = (Ax × Bx) + (Ay × By)

4. Magnitude (|V|): Also known as the length or Euclidean norm of the vector. It is calculated using the Pythagorean theorem based on the vector's components:

  • |V| = √(x² + y²)

Example Calculation

Suppose you have two forces acting on an object:

  • Vector A: (3, 4) — for example, 3 units East and 4 units North.
  • Vector B: (1, 2) — for example, 1 unit East and 2 units North.

Using the calculator:

  • Addition: (3+1, 4+2) = (4, 6)
  • Magnitude of A: √(3² + 4²) = √(9 + 16) = 5
  • Dot Product: (3×1) + (4×2) = 3 + 8 = 11

Applications in Physics

Vectors are used daily in various scenarios:

  • Navigation: Calculating the resultant speed and direction of a plane when affected by wind (wind velocity vector + plane velocity vector).
  • Structural Engineering: Summing force vectors to ensure a building or bridge is in equilibrium (Total Force = 0).
  • Game Development: Determining the movement of characters and how they bounce off surfaces using surface normals and dot products.

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