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Vector Calculator

Free online vector calculator with step-by-step solutions. Calculate dot product, cross product, magnitude, unit vector, angle between vectors, projection, and more with interactive 3D visualization.

Vector Calculator
Quick Examples
1 Enter Vectors
Components separated by commas
Required for two-vector operations
2 Select Operation

Embed Vector Calculator Widget

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.

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]:

Magnitude Formula
$$||A|| = \sqrt{a^2 + b^2 + c^2}$$

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:

Unit Vector Formula
$$\hat{A} = \frac{A}{||A||}$$

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:

Dot Product Formula
$$A \cdot B = a_1 b_1 + a_2 b_2 + a_3 b_3 = ||A|| \cdot ||B|| \cdot \cos(\theta)$$

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:

Cross Product Formula
$$A \times B = (a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1)$$

Vector Addition and Subtraction

Vector addition combines vectors by adding corresponding components. Subtraction finds the difference:

Addition/Subtraction
$$A + B = [a_1 + b_1, a_2 + b_2, a_3 + b_3]$$ $$A - B = [a_1 - b_1, a_2 - b_2, a_3 - b_3]$$

Angle Between Vectors

The angle between two vectors is found using the relationship between dot product and magnitudes:

Angle Formula
$$\theta = \arccos\left(\frac{A \cdot B}{||A|| \cdot ||B||}\right)$$

Vector Projection

The projection of vector A onto vector B gives the component of A in the direction of B:

Projection Formula
$$\text{proj}_B A = \frac{A \cdot B}{B \cdot B} \cdot B$$

Scalar Multiplication

Scalar multiplication multiplies each component of a vector by a number, scaling the vector:

Scalar Multiplication
$$k \cdot A = [k \cdot a_1, k \cdot a_2, k \cdot a_3]$$

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

  1. Enter Vector A: Input the components of your first vector, separated by commas (e.g., 3, 4, 0).
  2. Enter Vector B (if needed): For two-vector operations, input the second vector.
  3. Select Operation: Choose which calculation to perform from the dropdown menu.
  4. Set Precision: Choose how many decimal places you want in your results.
  5. 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

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

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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