Vector Calculator

Category: Linear Algebra
- April 16, 2025
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Calculate vector operations including addition, subtraction, dot product, cross product, and more. This calculator supports both 2D and 3D vectors and provides step-by-step calculation details.

Vector Inputs

Vector 1
Vector 2

Display Options

About Vector Operations

Vectors are mathematical objects that have both magnitude and direction. They are fundamental in physics, engineering, computer graphics, and many other fields. This calculator supports common vector operations in both 2D and 3D space.

Key Vector Operations
  • Addition: Adds corresponding components of two vectors, resulting in a new vector.
  • Subtraction: Subtracts the components of one vector from another, useful for finding displacement vectors.
  • Dot Product: A scalar product of two vectors that gives a single number, related to the angle between vectors.
  • Cross Product: Produces a vector perpendicular to both input vectors (3D only).
  • Magnitude: The length of a vector, calculated using the Pythagorean theorem.
  • Normalization: Creates a unit vector (length 1) pointing in the same direction as the original vector.
  • Angle Between Vectors: Calculated using the dot product formula.
  • Projection: Projects one vector onto another, useful in physics and graphics.
  • Scalar Multiplication: Multiplies a vector by a scalar, changing its magnitude but not direction.
Applications

Vector calculations are essential in numerous fields:

  • Physics (forces, velocities, fields)
  • Engineering (structural analysis, electrical circuits)
  • Computer graphics and game development
  • Robotics and navigation
  • Machine learning and data analysis
  • Geometry and linear algebra

Vector Addition:

$$ \vec{a} + \vec{b} = (a_x + b_x,\ a_y + b_y,\ a_z + b_z) $$

Dot Product:

$$ \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z $$

Cross Product:

$$ \vec{a} \times \vec{b} = (a_y b_z - a_z b_y,\ a_z b_x - a_x b_z,\ a_x b_y - a_y b_x) $$

Magnitude of a Vector:

$$ |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} $$

Angle Between Vectors:

$$ \theta = \cos^{-1} \left( \frac{ \vec{a} \cdot \vec{b} }{|\vec{a}| \cdot |\vec{b}|} \right) $$

Vector Projection:

$$ \text{proj}_{\vec{b}}\vec{a} = \left( \frac{ \vec{a} \cdot \vec{b} }{|\vec{b}|^2} \right) \vec{b} $$

Normalization:

$$ \hat{a} = \frac{ \vec{a} }{|\vec{a}|} $$

Scalar Multiplication:

$$ k\vec{a} = (k \cdot a_x,\ k \cdot a_y,\ k \cdot a_z) $$

What Is the Vector Calculator?

The Vector Calculator is an easy-to-use online tool that helps you perform essential vector operations in both 2D and 3D space. It covers calculations such as vector addition, subtraction, dot product, cross product, magnitude, normalization, angle between vectors, projection, and scalar multiplication. This makes it useful for students, educators, engineers, and anyone working with linear algebra or physics.

How to Use the Vector Calculator

  • Select the number of dimensions: 2D or 3D.
  • Choose the vector operation you want to perform from the dropdown menu.
  • Enter the vector components and scalar (if needed).
  • Use the options to set decimal precision, show calculation steps, or visualize the vectors.
  • Click the Calculate button to get your result.
  • Click Reset to start a new calculation.

Key Features

  • Step-by-step results: Clearly explains each stage of the calculation.
  • Supports 2D and 3D: Choose the dimensionality that fits your problem.
  • Interactive visualization: See your vectors and results on a live canvas.
  • Versatile operations: From simple vector addition to cross products and projections.
  • Precision control: Adjust decimal places to suit your needs.

Why Use This Calculator?

Vector math is essential in many areas of study and work. This calculator simplifies the process by letting you focus on the problem, not the math. Whether you're working on force analysis in physics, directional data in computer graphics, or solving problems in linear algebra, this tool gives accurate answers fast.

You can also pair it with other tools like the Vector Addition Tool to explore how vectors combine, or use a Vector Magnitude Tool to isolate vector length. When working on broader matrix-related problems, you may find tools such as the LU Decomposition Calculator, QR Factorization Tool, and Matrix Inverse Tool helpful.

Frequently Asked Questions (FAQ)

What is a vector?

A vector is a quantity that has both magnitude (length) and direction. Vectors are represented by coordinates in 2D or 3D space, such as (x, y) or (x, y, z).

Can I use this tool for physics problems?

Yes. It’s useful for solving problems involving force, velocity, displacement, and other vector-related quantities in physics.

What’s the difference between dot and cross product?

The dot product results in a scalar value and relates to the angle between two vectors. The cross product produces another vector that is perpendicular to the original two (only in 3D).

What if I only need to multiply a vector by a number?

Use the scalar multiplication option to scale a vector by a constant value (e.g., double its length).

Can I see how the calculation was done?

Yes. Check the “Show calculation steps” box before you calculate to view a detailed breakdown.

Related Tools You Might Find Useful

This Vector Calculator is a practical and reliable tool for anyone learning or working with vectors. It supports a wide range of operations and presents results in a way that’s both clear and informative.