A vector is a mathematical object that has both magnitude (length) and direction. In 2D, a vector is defined by two components (x, y), and in 3D by three components (x, y, z). Vectors are used extensively in physics, engineering, computer graphics, and machine learning.

How is vector magnitude calculated?

The magnitude (or length) of a vector v = (x, y, z) is sqrt(x^2 + y^2 + z^2). For 2D vectors v = (x, y), it is sqrt(x^2 + y^2). This is the Euclidean distance from the origin to the point represented by the vector.

What does normalizing a vector mean?

Normalizing a vector means scaling it to have a magnitude of 1 while preserving its direction. The unit vector is computed as v / |v|. Normalizing is useful when you only care about direction, not magnitude, such as in lighting calculations in computer graphics.

Vector Calculator - Dot Product, Cross Product,

Q: What is the dot product?

The dot product of two vectors a and b is a scalar value computed as a1*b1 + a2*b2 + a3*b3. It equals |a|*|b|*cos(theta), where theta is the angle between the vectors. A dot product of zero means the vectors are perpendicular.

Q: What is the cross product?

The cross product of two 3D vectors produces a new vector perpendicular to both input vectors. For a = (a1,a2,a3) and b = (b1,b2,b3), the cross product is (a2*b3-a3*b2, a3*b1-a1*b3, a1*b2-a2*b1). Its magnitude equals |a|*|b|*sin(theta), representing the area of the parallelogram formed by the two vectors.

Vector Calculator

Compute vector operations for 2D and 3D vectors: addition, subtraction, dot product, cross product, magnitude, normalization, and scalar multiplication.

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Vector Operations

Operation

Vector 1 (comma-separated)

Enter 2 or 3 components separated by commas.

Vector 2 (comma-separated)

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How to Use the Vector Calculator

Select an operation, enter vector components separated by commas (2D or 3D), and see results instantly. For binary operations, both vectors must have the same number of dimensions. Cross product is available for 3D vectors only.

For example, the cross product of ⟨1, 2, 3⟩ and ⟨4, 5, 6⟩ gives ⟨−3, 6, −3⟩. The dot product of the same vectors gives 1×4 + 2×5 + 3×6 = 32.

Vector Operations Explained

Addition and subtraction are performed component-wise: (a₁, a₂, a₃) + (b₁, b₂, b₃) = (a₁+b₁, a₂+b₂, a₃+b₃).

The dot product yields a scalar: a · b = |a||b|cosθ. It measures how much two vectors point in the same direction. Perpendicular vectors have a dot product of zero.

The cross product (3D only) yields a vector perpendicular to both inputs. Its magnitude equals the area of the parallelogram formed by the two vectors, and it follows the right-hand rule for direction.

Applications

Vectors are fundamental in physics (force, velocity, acceleration), computer graphics (3D rendering, lighting, transformations), machine learning (feature vectors, embeddings), navigation (GPS, flight paths), and engineering (stress analysis, fluid flow). Understanding vector operations is essential for any quantitative STEM field.

Frequently Asked Questions

What is a vector?

A mathematical object with both magnitude and direction, defined by 2 or 3 components in a coordinate system.

What is the dot product?

A scalar product of two vectors equalling |a||b|cosθ. Zero dot product means the vectors are perpendicular.

What is the cross product?

A 3D operation producing a vector perpendicular to both inputs with magnitude |a||b|sinθ.

How is magnitude calculated?

|v| = √(x² + y² + z²) for 3D, or √(x² + y²) for 2D.

What is normalization?

Scaling a vector to unit length (magnitude 1) while preserving direction: v/|v|.

Disclaimer: This calculator is for informational and educational purposes only. Results are estimates and should not be considered professional expert advice. Consult a qualified professional before making decisions based on these calculations. See our full Disclaimer.