Vector Calculator

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]:

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

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

• 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

• Vector (mathematics and physics) - Wikipedia
• Vectors and Spaces - Khan Academy
• Dot Product - Wikipedia
• Cross Product - Wikipedia

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