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Calculator Answers

Vector Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A vector calculator is a linear algebra tool that performs mathematical operations on vectors in two-dimensional or three-dimensional Cartesian coordinate space. The calculator supports operations including vector addition, subtraction, scalar multiplication, dot product, cross product, magnitude computation, and angle determination. It applies vector arithmetic formulas, such as multiplying corresponding elements and summing them for dot products, to ensure perfect precision. Physics and engineering students use this utility to analyze forces and spatial rotations.

Pick 2D or 3D, pick an operation, and the calculator returns the result. Supports the standard list of vector operations used in geometry, physics, and graphics.

Quick Answer

Perform vector algebra operations in 2D or 3D space. Enter vector coordinates to calculate addition, subtraction, dot product, cross product, and magnitude.

Dimension

Operation

Vector A

Vector B

Vector result

Dot product A · B

32

Dot product checks orthogonality (zero means perpendicular). Cross product gives a vector perpendicular to both inputs whose magnitude equals |A||B|sin θ.

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Examples

(1,2,3) · (4,5,6)

= 32

(1,0,0) × (0,1,0)

= (0, 0, 1)

|(3,4)|

= 5

angle((1,0), (0,1))

= 90°

How it works

Each operation is the standard one taught in linear algebra class. Component-wise for add/subtract/scalar; sums of products for dot; the right-hand-rule formula for cross; Pythagorean for magnitude; division by magnitude for unit vector.

Magnitude · |A| = √(x² + y² + z²)

Dot product · A · B = Σᵢ Aᵢ · Bᵢ

Cross product (3D) · A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

Angle · cos θ = (A · B) / (|A| · |B|)

Learn the concept

These guides explain what the vectors this tool adds are describing.

What Is a Translation in Math? A translation IS a vector: two slides compose by adding them.

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Cross Product CalculatorCalculate the cross product A × B of two 3D vectors, with step-by-step determinant method.

Frequently asked questions

A vector is an ordered list of numbers that represents a quantity with both magnitude and direction. In 2D it has x and y components; in 3D it has x, y, and z.

The dot product (also called inner product) returns a scalar. If A·B = 0, the vectors are perpendicular. The sign tells you whether the angle between them is acute (positive) or obtuse (negative).

Only for 3D vectors. The result is a vector perpendicular to both inputs, with magnitude equal to |A||B|sin θ. Use the right-hand rule to find the direction.

A unit vector is a vector of length 1 pointing in the same direction as the original. Computed as A / |A|. The zero vector has no unit vector (division by zero).

cos θ = (A · B) / (|A| · |B|). Take the inverse cosine to get the angle. The calculator returns both degrees and radians and shows the cos θ value used.

No. This calculator covers 2D and 3D, the common cases for geometry, physics, and graphics. For higher-dimensional vector arithmetic, use a CAS or a programming language.