Vector Calculator

Vector Operations Calculator

Input Vectors & Scalar (2D)

Perform Operations

function calculateVectorAddition() { var vectorAx = parseFloat(document.getElementById("vectorAx").value); var vectorAy = parseFloat(document.getElementById("vectorAy").value); var vectorBx = parseFloat(document.getElementById("vectorBx").value); var vectorBy = parseFloat(document.getElementById("vectorBy").value); if (isNaN(vectorAx) || isNaN(vectorAy) || isNaN(vectorBx) || isNaN(vectorBy)) { document.getElementById("resultAdd").innerHTML = "Please enter valid numbers for all vector components."; return; } var resultX = vectorAx + vectorBx; var resultY = vectorAy + vectorBy; document.getElementById("resultAdd").innerHTML = "(" + resultX.toFixed(2) + ", " + resultY.toFixed(2) + ")"; } function calculateVectorSubtraction() { var vectorAx = parseFloat(document.getElementById("vectorAx").value); var vectorAy = parseFloat(document.getElementById("vectorAy").value); var vectorBx = parseFloat(document.getElementById("vectorBx").value); var vectorBy = parseFloat(document.getElementById("vectorBy").value); if (isNaN(vectorAx) || isNaN(vectorAy) || isNaN(vectorBx) || isNaN(vectorBy)) { document.getElementById("resultSubtract").innerHTML = "Please enter valid numbers for all vector components."; return; } var resultX = vectorAx – vectorBx; var resultY = vectorAy – vectorBy; document.getElementById("resultSubtract").innerHTML = "(" + resultX.toFixed(2) + ", " + resultY.toFixed(2) + ")"; } function calculateScalarMultiplication() { var vectorAx = parseFloat(document.getElementById("vectorAx").value); var vectorAy = parseFloat(document.getElementById("vectorAy").value); var scalarK = parseFloat(document.getElementById("scalarK").value); if (isNaN(vectorAx) || isNaN(vectorAy) || isNaN(scalarK)) { document.getElementById("resultScalar").innerHTML = "Please enter valid numbers for Vector A components and scalar."; return; } var resultX = scalarK * vectorAx; var resultY = scalarK * vectorAy; document.getElementById("resultScalar").innerHTML = "(" + resultX.toFixed(2) + ", " + resultY.toFixed(2) + ")"; } function calculateMagnitude() { var vectorAx = parseFloat(document.getElementById("vectorAx").value); var vectorAy = parseFloat(document.getElementById("vectorAy").value); if (isNaN(vectorAx) || isNaN(vectorAy)) { document.getElementById("resultMagnitude").innerHTML = "Please enter valid numbers for Vector A components."; return; } var magnitude = Math.sqrt(vectorAx * vectorAx + vectorAy * vectorAy); document.getElementById("resultMagnitude").innerHTML = magnitude.toFixed(2); }

Understanding Vectors and Their Operations

Vectors are fundamental mathematical objects used in physics, engineering, computer graphics, and many other fields to represent quantities that have both magnitude (size) and direction. Unlike scalars, which only have magnitude (like temperature or mass), vectors provide a complete description of a directed quantity.

What is a Vector?

A vector can be visualized as an arrow pointing from one point to another. In a 2D or 3D coordinate system, a vector is often represented by its components along the axes. For example, a 2D vector A can be written as (Ax, Ay), where Ax is its component along the x-axis and Ay is its component along the y-axis.

Common examples of vector quantities include:

• Displacement: The change in position of an object.
• Velocity: The rate of change of displacement.
• Acceleration: The rate of change of velocity.
• Force: A push or pull on an object.

Key Vector Operations Explained

1. Vector Addition (A + B)

When you add two vectors, you are essentially finding the resultant vector that represents the combined effect of the two individual vectors. Geometrically, if you place the tail of vector B at the head of vector A, the sum A + B is the vector from the tail of A to the head of B (the "triangle rule").

Mathematically, vector addition is performed by adding their corresponding components:

If A = (Ax, Ay) and B = (Bx, By), then A + B = (Ax + Bx, Ay + By).

Example: If A = (3, 4) and B = (1, 2), then A + B = (3+1, 4+2) = (4, 6).

2. Vector Subtraction (A – B)

Vector subtraction can be thought of as adding the negative of a vector. The negative of a vector B, denoted as –B, has the same magnitude as B but points in the opposite direction. So, A – B = A + (-B).

Mathematically, vector subtraction is performed by subtracting their corresponding components:

If A = (Ax, Ay) and B = (Bx, By), then A – B = (Ax – Bx, Ay – By).

Example: If A = (3, 4) and B = (1, 2), then A – B = (3-1, 4-2) = (2, 2).

3. Scalar Multiplication (k * A)

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation changes the magnitude of the vector but not its direction (unless the scalar is negative, in which case it reverses the direction).

Mathematically, each component of the vector is multiplied by the scalar:

If A = (Ax, Ay) and k is a scalar, then k * A = (k * Ax, k * Ay).

Example: If A = (3, 4) and k = 2, then 2 * A = (2*3, 2*4) = (6, 8).

4. Magnitude of a Vector (|A|)

The magnitude of a vector represents its length or size. For a 2D vector, it can be calculated using the Pythagorean theorem, as the components form the sides of a right-angled triangle.

Mathematically, the magnitude of vector A = (Ax, Ay) is given by:

|A| = sqrt(Ax² + Ay²)

Example: If A = (3, 4), then |A| = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.

This calculator allows you to quickly perform these fundamental operations on 2D vectors, helping you understand and verify vector calculations for your studies or projects.