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Vector Calculator

Enter two vectors A and B in 2D or 3D and choose an operation. You get the answer with every step worked out using your numbers, plus a table with all the other operations at once and a diagram for 2D vectors.

Dimension

2D (x, y)3D (x, y, z)

Operation

Vector A

A: x component

A: y component

A: z component

Vector B

B: x component

B: y component

B: z component

Scalar k

Enter your values to see the result.

How to use the Vector Calculator

Choose 2D or 3D.
Pick the operation, for example the cross product or the angle between the vectors.
Enter the components of vector A and vector B.
Press Calculate to see the result and the steps.
Check the table for every other operation, or copy the results.

Vector formulas

Operation	Formula (A = ⟨Aₓ, Aᵧ, A𝓏⟩, B = ⟨Bₓ, Bᵧ, B𝓏⟩)	Result type
Addition	A + B = ⟨Aₓ + Bₓ, Aᵧ + Bᵧ, A𝓏 + B𝓏⟩	Vector
Subtraction	A − B = ⟨Aₓ − Bₓ, Aᵧ − Bᵧ, A𝓏 − B𝓏⟩	Vector
Scalar multiple	kA = ⟨kAₓ, kAᵧ, kA𝓏⟩	Vector
Magnitude	\|A\| = √(Aₓ² + Aᵧ² + A𝓏²)	Scalar
Dot product	A · B = AₓBₓ + AᵧBᵧ + A𝓏B𝓏 = \|A\|\|B\| cos θ	Scalar
Cross product	A × B = ⟨AᵧB𝓏 − A𝓏Bᵧ, A𝓏Bₓ − AₓB𝓏, AₓBᵧ − AᵧBₓ⟩	Vector (3D)
Angle	θ = arccos((A · B) / (\|A\|\|B\|))	0° to 180°
Unit vector	Â = A / \|A\|	Vector of length 1
Vector projection	proj_B A = (A · B / \|B\|²) B	Vector
Scalar projection	comp_B A = A · B / \|B\|	Scalar

Worked example

For A = ⟨3, −2, 5⟩ and B = ⟨1, 4, −1⟩:

A · B = 3·1 + (−2)·4 + 5·(−1) = −10, so the angle is obtuse.
|A| = √38 ≈ 6.1644, |B| = √18 ≈ 4.2426, so cos θ = −10 / 26.153 ≈ −0.3824 and θ ≈ 112.48°.
A × B = ⟨(−2)(−1) − 5·4, 5·1 − 3·(−1), 3·4 − (−2)·1⟩ = ⟨−18, 8, 14⟩, with length √584 ≈ 24.166 (the parallelogram area).

Properties worth remembering

The dot product is commutative (A · B = B · A); the cross product is anti-commutative (A × B = −(B × A)).
A · B = 0 means perpendicular; A × B = 0 means parallel (or a zero vector).
|A × B| = |A||B| sin θ, the area of the parallelogram spanned by A and B; half of it is the triangle area.
The direction of A × B follows the right-hand rule.

Frequently asked questions

How do you calculate the dot product?

Multiply matching components and add them: A · B = AₓBₓ + AᵧBᵧ + A𝓏B𝓏. For ⟨3, −2, 5⟩ and ⟨1, 4, −1⟩ that is 3 − 8 − 5 = −10. A dot product of 0 means the vectors are perpendicular.

How do you calculate the cross product?

For 3D vectors, A × B = ⟨AᵧB𝓏 − A𝓏Bᵧ, A𝓏Bₓ − AₓB𝓏, AₓBᵧ − AᵧBₓ⟩. The result is perpendicular to both A and B, and its length equals the area of the parallelogram they span.

Is there a cross product in 2D?

Not as a vector, but treating 2D vectors as ⟨x, y, 0⟩ gives ⟨0, 0, AₓBᵧ − AᵧBₓ⟩. The scalar AₓBᵧ − AᵧBₓ is often called the 2D cross product: its sign tells whether B is counter-clockwise (positive) or clockwise (negative) from A.

How do I find the angle between two vectors?

Use cos θ = (A · B) / (|A| |B|), then θ = arccos of that value. The angle is always between 0° and 180°. It is undefined if either vector is the zero vector.

What is a unit vector?

A vector of length 1 pointing in the same direction. Divide each component by the magnitude: Â = A / |A|. For ⟨3, 4⟩, |A| = 5 and Â = ⟨0.6, 0.8⟩.

What is the projection of A onto B?

The vector projection is proj_B A = (A · B / |B|²) B, the shadow of A along B. The scalar projection A · B / |B| is its signed length. A − proj_B A is the part of A perpendicular to B.

How is the magnitude of a vector calculated?

With the Pythagorean theorem: |A| = √(Aₓ² + Aᵧ² + A𝓏²). For ⟨3, −2, 5⟩ that is √38 ≈ 6.164414.

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