What is a vector calculator?

A vector calculator is a tool that computes mathematical operations on two-dimensional or three-dimensional vectors. It takes the coordinates of vectors u and v and calculates their sum, difference, dot product, cross product, magnitudes, unit vectors, projection, and the angle between them. Our vector calculator runs entirely in your browser with no signup required.

How do you calculate the dot product of two vectors?

The dot product of two vectors u and v is calculated by multiplying corresponding components and adding the results: u · v = ux*vx + uy*vy + uz*vz. It is a scalar value that represents the product of their magnitudes and the cosine of the angle between them. If the dot product is zero, the vectors are orthogonal (perpendicular).

What is the cross product, and why is it only for 3D vectors?

The cross product is a vector operation that produces a third vector perpendicular to both input vectors. It is defined specifically in three-dimensional space because it requires a third dimension to represent the perpendicular output. For two 2D vectors, the cross product magnitude can be calculated, which corresponds to the z-component of their 3D equivalents.

How is the projection of a vector calculated?

The projection of vector u onto vector v (denoted proj_v u) is the orthogonal projection of u onto the line spanned by v. The formula is: proj_v u = ((u · v) / ||v||²) * v. This results in a vector pointing in the direction of v (or opposite, if the dot product is negative) with a magnitude equal to the component of u along v.

Does this vector calculator support fractions or decimals?

Yes. Our vector calculator supports integers, decimals (e.g. 2.5), and fractional formats (e.g. 1/2 or -3/4) for all coordinates. The calculator automatically parses fractions into numerical values to ensure accurate computations and clean step-by-step logs.

Is my data safe when using this vector calculator?

Yes. The vector calculator operates entirely client-side in your browser. All coordinate inputs, math operations, and graph renderings are calculated on your local device. No data is sent to our servers, ensuring 100% privacy and security.

Vector Calculator

Analyze and visualize 2D and 3D vectors instantly. Enter coordinates for vectors u and v to calculate their sum, difference, dot product, cross product, magnitudes, unit vectors, projection, and the angle between them. Review complete step-by-step arithmetic steps alongside an interactive visual coordinate graph with perspective rotation control. All calculations are performed 100% locally in your browser.

Vector Computation & Graphing

Input coordinates in standard format, select dimension mode, and view computed linear algebra parameters. Hover or click result items to review steps.

Quick Presets

Vector u Input

u_x (horizontal)

u_y (vertical)

Vector v Input

v_x (horizontal)

v_y (vertical)

Scalar Multiplier (c)

Vector Space Graph

Plot Resultant:

Vectors represented: u, v, and resultant dashed. Graph scales dynamically.

Calculated Properties & Equations

Magnitudes (Lengths)

||u||5

√(3² + 4² )

||v||2.2361

√(1² + 2² )

Dot Product (Scalar)

u · v11

u · v = u_xv_x + u_yv_y + u_zv_z

= (3)·(1) + (4)·(2)

= 3 + 8

Angle Between Vectors

θ (Degrees)10.3048°

θ (Radians)0.1799 rad

cos(θ) = (u · v) / (||u|| * ||v||)

= 11 / (5 * 2.2361)

= 0.9839

Sum & Difference

u + v (Sum)(4, 6)

(3 + 1, 4 + 2)

u - v (Difference)(2, 2)

(3 - 1, 4 - 2)

Scalar Multiplications (c = 2)

c · u(6, 8)

(2 × 3, 2 × 4)

c · v(2, 4)

(2 × 1, 2 × 2)

Cross Product (Vector)

u × v (z-component magnitude)2

u_xv_y - u_yv_x = (3)(2) - (4)(1)

Unit Vectors (Directional)

û (Unit u)(0.6, 0.8)

u / ||u|| = (3/5, 4/5)

v̂ (Unit v)(0.4472, 0.8944)

v / ||v|| = (1/2.2361, 2/2.2361)

Orthogonal Projections

proj_v u(2.2, 4.4)

((u · v) / ||v||²) · v = (2.2) · v

proj_u v(1.32, 1.76)

((u · v) / ||u||²) · u = (0.44) · u

Why Use Our Vector Calculator?

Instant 2D & 3D Computation

Perform complex vector addition, subtraction, dot product, cross product, angles, and projections in real-time. The vector calculator processes your inputs immediately inside your browser with no delay.

Secure Vector Calculator Online

The vector calculator executes entirely client-side on your local device. Your coordinate coordinates and mathematical expressions are never sent to a server, guaranteeing complete security and privacy.

Vector Graphing & Visualizations

Render vectors u and v alongside their resultants dynamically. Switch between a 2D coordinate grid and a 3D perspective projection complete with adjustable rotation angle sliders for depth perception.

Detailed Step-by-Step Logic

Learn linear algebra concepts through detailed step-by-step explanations. Our tool displays the formulas, intermediate coordinate multiplications, and final additions for every operation.

Common Use Cases for Vector Calculator

Linear Algebra Coursework

Students use the vector calculator to verify hand-solved assignments for linear algebra, calculus, and multi-variable mathematics. Instantly cross-check dot/cross products, magnitudes, and angle calculations.

Physics Mechanics & Dynamics

Engineers and physics students model forces, velocity vectors, and rotational torque. Compute resultant forces by adding 2D/3D component vectors and calculate perpendicular torque vectors using cross products.

3D Game Development & Rendering

Developers calculate camera directions, lighting angles, and surface normal vectors. The vector calculator helps verify calculations for dot products (used in backface culling and diffuse lighting) and cross products.

Computer Graphics Algorithms

Use the vector calculator to study coordinate geometry transformations. Determine projection vectors to projects coordinates onto arbitrary planes, and solve angles to calculate reflection rays.

Navigation and Flight Planning

Navigate wind correction angles and flight paths. Model airspeed and wind vectors to calculate the ground speed and true heading using vector addition and angular trigonometry.

Coordinate Geometry Reference

Verify geometric properties such as parallelism, perpendicularity, and collinearity. Check if the angle is 0, 90, or 180 degrees, or if the dot product equals zero.

Understanding Vector Mathematics

What is a Vector?

A vector is a mathematical object that has both a magnitude (length) and a direction. In coordinate geometry, a vector is represented by its components along the coordinate axes: u = (u_x, u_y) for 2D, and u = (u_x, u_y, u_z) for 3D. Vectors are foundational in physics and computer science for representing velocity, forces, acceleration, and spatial orientation. Our vector calculator helps you perform and visualize coordinate calculations instantly.

How Our Vector Calculator Works

Select Dimension and Enter Coordinates: Choose between 2D and 3D vector modes. Input integers, decimals, or fractions (such as 1/2 or -3/4) into the fields for vector u and vector v.
Review Calculated Results: The vector calculator computes vector sums, differences, magnitudes, dot products, cross products, angles, and projections in real-time inside your browser.
Visualize and Study Steps: View the coordinates represented as directed arrows on a responsive grid (2D Cartesian coordinate plane or 3D perspective plot), and review the detailed step-by-step arithmetic explanations for each output.

Core Vector Operations and Formulas

Our tool handles the following vector operations using standard algebraic formulas:

Vector Addition: Adding corresponding components: (u_x + v_x, u_y + v_y, u_z + v_z).
Dot Product:A scalar operation representing coordinate multiplication: u · v = u_xv_x + u_yv_y + u_zv_z.
Cross Product (3D only):Produces a vector orthogonal to both inputs: u × v = (u_yv_z - u_zv_y, u_zv_x - u_xv_z, u_xv_y - u_yv_x).
Vector Magnitude:The Cartesian distance or length of the vector: ||u|| = √(u_x² + u_y² + u_z²).
Vector Projection: The orthogonal shadow of vector u cast onto vector v: proj_vu = ((u · v) / ||v||²) · v.
Angle Between Vectors:Derived from the dot product relationship: cos(θ) = (u · v) / (||u|| · ||v||), yielding θ = arccos(cos(θ)).

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Frequently Asked Questions About Vector Calculator

A vector calculator is an online tool designed to calculate basic algebraic operations on 2D and 3D vectors. By entering vector coordinates, you can compute addition, subtraction, dot products, cross products, magnitudes, unit vectors, projections, and angles with visual coordinate graphs.

The dot product (scalar product) multiplies corresponding components and adds them to produce a single number, reflecting how much two vectors point in the same direction. The cross product (vector product), which is only defined in 3D, outputs a new vector that is perpendicular to both input vectors, representing the rotational axis and area spanned by them.

Yes. The vector calculator executes entirely client-side on your local device. None of your coordinates, formulas, or calculations are transmitted to any external server or saved in a database, ensuring 100% privacy and confidentiality.

Yes, our vector calculator is 100% free with no hidden charges, premium features, signups, or usage limits. You can perform calculations and visualize vectors in 2D and 3D as much as you need.

If either vector is a zero vector (0, 0) or (0, 0, 0), some calculations like unit vectors, projections, and angles are undefined because they require division by the vector magnitude. The calculator detects zero vectors and displays helpful warnings instead of crash errors.

The calculator computes the projection of u onto v and v onto u, showing the exact coordinate equations and components. The interactive SVG grid plots the projected vector along the axis of the target vector, helping you visualize the orthogonal component geometrically.

Yes. The calculator parses integers, floating-point decimals, and fractions like "1/2", "-3/5", or "7/4". This lets you input precise coordinate systems without rounding them to decimals beforehand.

The angle is calculated using the formula theta = arccos((u · v) / (||u|| * ||v||)). The resulting angle is shown in both degrees and radians. If the angle is 90 degrees (or pi/2 radians), the dot product is zero, confirming the vectors are perpendicular.

The 3D graph plots three-dimensional coordinates onto a 2D canvas using an isometric perspective projection. It renders X (red), Y (green), and Z (blue) axes. An adjustable rotation slider lets you spin the coordinates to look at the spatial layout from different angles.