Vector Addition Calculator

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Vector Addition Calculator

Vector 1

Vector 2

function calculateVectorAddition() { var vector1X = parseFloat(document.getElementById('vector1X').value); var vector1Y = parseFloat(document.getElementById('vector1Y').value); var vector2X = parseFloat(document.getElementById('vector2X').value); var vector2Y = parseFloat(document.getElementById('vector2Y').value); var resultDiv = document.getElementById('result'); if (isNaN(vector1X) || isNaN(vector1Y) || isNaN(vector2X) || isNaN(vector2Y)) { resultDiv.style.display = 'block'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Please enter valid numbers for all vector components.'; return; } var resultantX = vector1X + vector2X; var resultantY = vector1Y + vector2Y; var resultantMagnitude = Math.sqrt(resultantX * resultantX + resultantY * resultantY); var resultantAngleRad = Math.atan2(resultantY, resultantX); var resultantAngleDeg = resultantAngleRad * (180 / Math.PI); // Normalize angle to 0-360 degrees if (resultantAngleDeg < 0) { resultantAngleDeg += 360; } resultDiv.style.display = 'block'; resultDiv.style.backgroundColor = '#d4edda'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = '

Resultant Vector:

' + 'X-Component: ' + resultantX.toFixed(2) + " + 'Y-Component: ' + resultantY.toFixed(2) + " + 'Magnitude: ' + resultantMagnitude.toFixed(2) + " + 'Direction (Angle from positive X-axis): ' + resultantAngleDeg.toFixed(2) + '°'; }

Understanding Vector Addition

Vectors are fundamental quantities in physics, engineering, and mathematics, representing both magnitude (size) and direction. Unlike scalar quantities (like temperature or mass) which only have magnitude, vectors provide a complete description of a physical quantity. Common examples include displacement, velocity, force, and acceleration.

What is a Vector?

A vector can be visualized as an arrow pointing in a specific direction, with its length representing its magnitude. In a 2D coordinate system, a vector can be described by its components along the X and Y axes, often written as (X, Y).

  • Magnitude: The length of the vector. For a vector (X, Y), the magnitude is calculated as √(X² + Y²).
  • Direction: The angle the vector makes with a reference axis (usually the positive X-axis), measured counter-clockwise.

Why Add Vectors?

Vector addition is crucial for understanding how multiple forces, velocities, or displacements combine to produce a single, overall effect. For instance:

  • Physics: Calculating the net force on an object when multiple forces are acting on it.
  • Navigation: Determining the actual path and speed of an aircraft or boat when considering both its own velocity and the wind/current velocity.
  • Engineering: Analyzing stresses and strains in structures.
  • Computer Graphics: Combining transformations or movements of objects in a virtual environment.

How to Add Vectors (Component Method)

The simplest and most common method for adding vectors is the component method. If you have two vectors, A = (Ax, Ay) and B = (Bx, By), their resultant vector R is found by adding their corresponding components:

Rx = Ax + Bx
Ry = Ay + By

So, the resultant vector is R = (Rx, Ry).

Finding the Magnitude and Direction of the Resultant Vector

Once you have the components of the resultant vector (Rx, Ry), you can find its magnitude and direction:

  • Magnitude (||R||): This is the length of the resultant vector.

    ||R|| = √(Rx² + Ry²)

  • Direction (θ): This is the angle the resultant vector makes with the positive X-axis. It's typically calculated using the arctangent function.

    θ = arctan(Ry / Rx)

    It's important to use the atan2(y, x) function (available in most programming languages) to correctly determine the angle in all four quadrants, as atan(y/x) alone only gives angles in the first and fourth quadrants.

Example Calculation

Let's use the default values in the calculator:

  • Vector 1 (A): (3, 4)
  • Vector 2 (B): (2, -1)

1. Add the X-components:
Rx = Ax + Bx = 3 + 2 = 5

2. Add the Y-components:
Ry = Ay + By = 4 + (-1) = 3

So, the Resultant Vector (R) = (5, 3).

3. Calculate the Magnitude:
||R|| = √(5² + 3²) = √(25 + 9) = √34 ≈ 5.83

4. Calculate the Direction:
θ = atan2(3, 5) ≈ 0.5404 radians ≈ 30.96°

This calculator simplifies the process, allowing you to quickly find the resultant vector, its magnitude, and its direction for any two 2D vectors.

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