Vector Calculator
MiniWebtool’s Vector Calculator is a free online tool with step-by-step solutions for dot product, cross product, magnitude, unit vector, angle between vectors, projection, and more, with interactive 3D visualization.
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Enter two vectors and pick an operation below to get the sum, difference, dot or cross product, angle, projection and more, with the steps.
About Vector Calculator
Welcome to our Vector Calculator, a comprehensive tool for performing vector operations with detailed step-by-step solutions. Whether you are a student learning linear algebra, an engineer working with forces and velocities, or anyone who needs to compute vector mathematics, this calculator provides accurate results with clear explanations.
At a Glance
Vector Calculator performs vector operations on 2D and 3D vectors and shows the working step by step. Operations include magnitude, unit vector, dot and cross product, addition, subtraction, angle, projection and scalar multiplication.
| Feature | What you get | Good for |
|---|---|---|
| One-vector operations | Magnitude (length), unit vector and scalar multiplication, which multiplies each component by a number. | Normalizing a direction or scaling a vector |
| Dot product | Returns a scalar from two vectors; a result of zero means the vectors are perpendicular. | Testing for perpendicular vectors or measuring similarity |
| Cross product | Returns a new vector perpendicular to two 3D vectors; its magnitude equals the parallelogram area they form. | Finding a normal to a plane or an area |
| Angle and projection | Finds the angle between two vectors in degrees and the projection of A onto B. | Resolving a force into components along a direction |
| Add and subtract | Adds or subtracts matching components of two vectors. | Combining forces or finding a relative position |
| Input and precision | Enter components separated by commas, such as 3, 4, 0, and choose the number of decimal places. | Quick checks of linear algebra homework |
| Steps and 3D view | Each result comes with a step-by-step solution and an interactive vector visualization. | Learning how each formula is applied |
What is a Vector?
A vector is a mathematical object that has both magnitude (length) and direction. Vectors are typically represented as ordered lists of numbers called components. For example, a 3D vector might be written as [3, 4, 5] representing movement of 3 units along the x-axis, 4 units along the y-axis, and 5 units along the z-axis.
Vectors are fundamental in physics (representing forces, velocities, accelerations), computer graphics (3D transformations, lighting), machine learning (feature vectors, embeddings), and many other fields.
Supported Vector Operations
Magnitude (Length)
The magnitude of a vector, also called its length or norm, measures how long the vector is. For vector A = [a, b, c]:
Unit Vector
A unit vector has magnitude 1 and points in the same direction as the original vector. It is calculated by dividing each component by the magnitude:
Dot Product (Scalar Product)
The dot product of two vectors produces a scalar (single number). It measures how much one vector goes in the direction of another:
Key properties: If the dot product is zero, the vectors are perpendicular. Positive means they point in similar directions; negative means opposite.
Cross Product (Vector Product)
The cross product of two 3D vectors produces a new vector perpendicular to both inputs. The magnitude equals the area of the parallelogram formed by the vectors:
Vector Addition and Subtraction
Vector addition combines vectors by adding corresponding components. Subtraction finds the difference:
Angle Between Vectors
The angle between two vectors is found using the relationship between dot product and magnitudes:
Vector Projection
The projection of vector A onto vector B gives the component of A in the direction of B:
Scalar Multiplication
Scalar multiplication multiplies each component of a vector by a number, scaling the vector:
Operations Summary Table
| Operation | Input Required | Output Type | Common Uses |
|---|---|---|---|
| Magnitude | One vector | Scalar | Finding distance, normalizing vectors |
| Unit Vector | One vector | Vector | Direction representation, normalization |
| Dot Product | Two vectors | Scalar | Angle calculation, projection, similarity |
| Cross Product | Two 3D vectors | Vector | Finding perpendicular vectors, area calculation |
| Addition | Two vectors | Vector | Combining forces, displacement |
| Subtraction | Two vectors | Vector | Finding relative position, difference |
| Angle | Two vectors | Scalar (degrees) | Orientation, similarity measurement |
| Projection | Two vectors | Vector | Shadow calculations, component decomposition |
| Scalar Multiply | One vector + scalar | Vector | Scaling, resizing vectors |
How to Use This Calculator
- Enter Vector A: Input the components of your first vector, separated by commas (e.g.,
3, 4, 0). - Enter Vector B (if needed): For two-vector operations, input the second vector.
- Select Operation: Choose which calculation to perform from the dropdown menu.
- Set Precision: Choose how many decimal places you want in your results.
- Calculate: Click the button to see results with step-by-step explanations.
Frequently Asked Questions
What is a dot product?
The dot product (also called scalar product or inner product) of two vectors A and B is a scalar value calculated by multiplying corresponding components and summing the results: A·B = a₁b₁ + a₂b₂ + a₃b₃. It equals |A||B|cos(θ) where θ is the angle between vectors. A dot product of zero means vectors are perpendicular.
What is a cross product?
The cross product (also called vector product) of two 3D vectors A and B produces a new vector perpendicular to both input vectors. It is calculated using A×B = (a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁). The magnitude |A×B| equals the area of the parallelogram formed by A and B.
How do you calculate vector magnitude?
Vector magnitude (length) is calculated using the Euclidean norm: |A| = √(a₁² + a₂² + a₃²) for a 3D vector. This formula extends to any dimension by summing the squares of all components and taking the square root.
What is a unit vector?
A unit vector is a vector with magnitude 1 that points in the same direction as the original vector. It is calculated by dividing each component by the vector's magnitude: Â = A/|A|. Unit vectors are useful for representing directions without magnitude.
How do you find the angle between two vectors?
The angle θ between vectors A and B is found using the dot product formula: cos(θ) = (A·B)/(|A||B|). Take the inverse cosine (arccos) of this value to get the angle in radians, then convert to degrees if needed by multiplying by 180/π.
What is vector projection?
Vector projection of A onto B gives the component of A in the direction of B. The formula is proj_B(A) = ((A·B)/(B·B)) × B. The scalar projection (component) is (A·B)/|B|. This is useful in physics for decomposing forces and velocities.
Applications of Vector Mathematics
- Physics: Representing forces, velocities, accelerations, electric and magnetic fields
- Computer Graphics: 3D transformations, lighting calculations, ray tracing
- Engineering: Structural analysis, fluid dynamics, robotics
- Machine Learning: Feature vectors, word embeddings, similarity measures
- Game Development: Character movement, collision detection, physics simulation
- Navigation: GPS calculations, flight paths, maritime routing
Additional Resources
Reference this content, page, or tool as:
"Vector Calculator" at https://MiniWebtool.com/vector-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Jan 27, 2026
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