What is a dot product?

The dot product (scalar product) of two vectors A and B is: A·B = ax×bx + ay×by + az×bz. The result is a single number (scalar). Geometrically, A·B = |A||B|cos(θ), where θ is the angle between the vectors. If A·B = 0, the vectors are perpendicular. If A·B > 0, the angle is acute; if A·B < 0, the angle is obtuse.

What is a cross product?

The cross product of 3D vectors A×B produces a new vector perpendicular to both A and B. Formula: A×B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx). The magnitude |A×B| = |A||B|sin(θ) gives the area of the parallelogram formed by the two vectors. Cross products only exist in 3D (and 7D).

How do you find the magnitude of a vector?

The magnitude (length) of a vector A = (ax, ay, az) is |A| = √(ax² + ay² + az²). In 2D: |A| = √(ax² + ay²). Magnitude is always a non-negative scalar.

What is a unit vector?

A unit vector has a magnitude of exactly 1 and points in the same direction as the original vector. To find the unit vector: Â = A / |A| = (ax/|A|, ay/|A|, az/|A|). Unit vectors are used to represent direction without magnitude, and are commonly denoted with a hat (î, ĵ, k̂ for the standard basis vectors).

Vector Calculator

Perform all common vector operations in 2D and 3D: addition, subtraction, dot product, cross product, magnitude, unit vector, and angle between vectors — with full step-by-step working.

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Enter Vectors

Examples:

Vector A

x

y

z

Vector B

x

y

z

Results

|A|

—

|B|

—

A·B

—

Angle

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A + B—

A − B—

B − A—

Dot product A·B—

Cross product A×B—

|A| (magnitude of A)—

|B| (magnitude of B)—

Â (unit vector of A)—

B̂ (unit vector of B)—

Angle θ between A and B—

Are A and B perpendicular?—

Are A and B parallel?—

Step-by-Step Working

Vector Formulas Reference

Operation	Formula (3D)	Result Type
Addition	A+B = (ax+bx, ay+by, az+bz)	Vector
Subtraction	A−B = (ax−bx, ay−by, az−bz)	Vector
Magnitude	\|A\| = √(ax²+ay²+az²)	Scalar
Unit vector	Â = A/\|A\|	Vector
Dot product	A·B = ax·bx + ay·by + az·bz	Scalar
Cross product	A×B = (ay·bz−az·by, az·bx−ax·bz, ax·by−ay·bx)	Vector (3D only)
Angle between	θ = arccos(A·B / (\|A\|\|B\|))	Angle (degrees)
Perpendicular test	A·B = 0	Boolean
Parallel test	\|A×B\| = 0 (or A = k·B)	Boolean

Frequently Asked Questions

The dot product (A·B) produces a scalar — a single number. It equals |A||B|cos(θ) and measures how much one vector projects onto another. The cross product (A×B) produces a new vector perpendicular to both A and B. Its magnitude equals |A||B|sin(θ), equal to the area of the parallelogram formed by the vectors. Cross products only exist in 3D.

Two vectors are perpendicular (orthogonal) when their dot product equals zero: A·B = 0. This is because A·B = |A||B|cos(θ), and cos(90°) = 0. Example: A = (3, 4) and B = (−4, 3) → A·B = 3×(−4) + 4×3 = −12 + 12 = 0 ✓.

Use the dot product formula: θ = arccos(A·B / (|A| × |B|)). This always gives the angle between 0° and 180°. Steps: 1) Calculate A·B. 2) Calculate |A| and |B|. 3) Divide: (A·B) / (|A||B|). 4) Take the inverse cosine.

A unit vector has magnitude 1 and represents pure direction without scale. To find it: divide every component by the original magnitude. Unit vectors are used in physics to represent directions (force direction, velocity direction), in graphics for surface normals, and in maths to separate the "direction" from the "magnitude" of a vector.