Vectors are fundamental mathematical objects used to represent quantities that have both magnitude (size) and direction. They are commonly used in physics, engineering, computer graphics, and many other scientific fields. This calculator allows you to perform several common vector operations: Addition, Subtraction, Dot Product, Cross Product, Scalar Multiplication, and Magnitude calculation.
Vector Components
A vector can be represented by its components along coordinate axes. For example, a 2D vector **A** can be written as (Ax, Ay) and a 3D vector **A** as (Ax, Ay, Az).
Operations Explained
Addition/Subtraction: To add or subtract two vectors, you add or subtract their corresponding components.
For 2D: **A** + **B** = (Ax + Bx, Ay + By)
For 3D: **A** + **B** = (Ax + Bx, Ay + By, Az + Bz)
Subtraction follows the same pattern: **A** – **B** = (Ax – Bx, Ay – By, Az – Bz).
Dot Product: The dot product (or scalar product) of two vectors results in a scalar (a single number). It is calculated by multiplying corresponding components and summing the results.
For 2D: **A** · **B** = Ax * Bx + Ay * By
For 3D: **A** · **B** = Ax * Bx + Ay * By + Az * Bz
The dot product is useful for finding the angle between two vectors and for projections.
Cross Product: The cross product (or vector product) is defined only for 3D vectors and results in a new vector that is perpendicular to both original vectors.
The direction of the resulting vector is determined by the right-hand rule.
Scalar Multiplication: To multiply a vector by a scalar (a single number), you multiply each component of the vector by that scalar.
k * **A** = (k*Ax, k*Ay, k*Az)
Magnitude: The magnitude (or length) of a vector is calculated using the Pythagorean theorem.
For 2D: |**A**| = sqrt(Ax^2 + Ay^2)
For 3D: |**A**| = sqrt(Ax^2 + Ay^2 + Az^2)
When to Use This Calculator
This calculator is ideal for students learning linear algebra and calculus, engineers solving problems involving forces or velocities, physicists analyzing motion, and programmers working with game development or computer graphics where vector math is essential.
function getFloat(id) {
var value = parseFloat(document.getElementById(id).value);
return isNaN(value) ? 0 : value;
}
function updateVisibility() {
var operation = document.getElementById('operation').value;
// Show/hide Z components based on 3D operations
var show3D = (operation === 'add' || operation === 'subtract' || operation === 'dot' || operation === 'cross' || operation === 'magnitude');
document.getElementById('Az_input_group').style.display = show3D ? " : 'none';
document.getElementById('Bz_input_group').style.display = show3D ? " : 'none';
// Show/hide Vector B inputs
var needsVectorB = (operation === 'add' || operation === 'subtract' || operation === 'dot' || operation === 'cross');
document.getElementById('vectorB_inputs').style.display = needsVectorB ? " : 'none';
// Show/hide Scalar input
var needsScalar = (operation === 'scalar_multiply' || operation === 'magnitude');
document.getElementById('scalar_input').style.display = needsScalar ? " : 'none';
// For magnitude, we only need Vector A
if (operation === 'magnitude') {
document.getElementById('vectorB_inputs').style.display = 'none';
}
}
document.getElementById('operation').addEventListener('change', updateVisibility);
updateVisibility(); // Set initial visibility
function calculateVector() {
var operation = document.getElementById('operation').value;
var resultDiv = document.getElementById('result');
var resultHTML = ";
var Ax = getFloat('Ax');
var Ay = getFloat('Ay');
var Az = getFloat('Az'); // Will be 0 if not visible or invalid
var Bx = getFloat('Bx');
var By = getFloat('By');
var Bz = getFloat('Bz'); // Will be 0 if not visible or invalid
var scalar = getFloat('scalarValue');
var resultVector = {};
try {
if (operation === 'add') {
var Cx = Ax + Bx;
var Cy = Ay + By;
var Cz = Az + Bz;
resultVector = { x: Cx, y: Cy, z: Cz };
resultHTML = "Result (A + B): (" + Cx.toFixed(4) + ", " + Cy.toFixed(4) + ", " + Cz.toFixed(4) + ")";
} else if (operation === 'subtract') {
var Cx = Ax – Bx;
var Cy = Ay – By;
var Cz = Az – Bz;
resultVector = { x: Cx, y: Cy, z: Cz };
resultHTML = "Result (A – B): (" + Cx.toFixed(4) + ", " + Cy.toFixed(4) + ", " + Cz.toFixed(4) + ")";
} else if (operation === 'dot') {
var dotProduct = Ax * Bx + Ay * By + Az * Bz;
resultHTML = "Dot Product (A · B): " + dotProduct.toFixed(4);
} else if (operation === 'cross') {
var Cx = Ay * Bz – Az * By;
var Cy = Az * Bx – Ax * Bz;
var Cz = Ax * By – Ay * Bx;
resultVector = { x: Cx, y: Cy, z: Cz };
resultHTML = "Cross Product (A x B): (" + Cx.toFixed(4) + ", " + Cy.toFixed(4) + ", " + Cz.toFixed(4) + ")";
} else if (operation === 'scalar_multiply') {
var kAx = scalar * Ax;
var kAy = scalar * Ay;
var kAz = scalar * Az;
resultVector = { x: kAx, y: kAy, z: kAz };
resultHTML = "Result (scalar * A): (" + kAx.toFixed(4) + ", " + kAy.toFixed(4) + ", " + kAz.toFixed(4) + ")";
} else if (operation === 'magnitude') {
var magnitude = Math.sqrt(Math.pow(Ax, 2) + Math.pow(Ay, 2) + Math.pow(Az, 2));
resultHTML = "Magnitude (|A|): " + magnitude.toFixed(4);
} else {
resultHTML = "Please select a valid operation.";
}
resultDiv.innerHTML = resultHTML;
} catch (e) {
resultDiv.innerHTML = "Error: " + e.message + "";
}
}