Vector Calculator

3D vector add/subtract, dot & cross product, magnitude, angle

Quick Answer

Vector Calculator: 3D vector add/subtract, dot & cross product, magnitude, angle. Add/subtract: a ± b = (a₁±b₁, a₂±b₂, a₃±b₃).

Vector a

Vector b

a + b(5.000, 7.000, 9.000)

a − b(-3.000, -3.000, -3.000)

a · b32.0000

|a|3.7417

|b|8.7750

Angle θ12.93°

a × b（cross product）

(-3.000, 6.000, -3.000)

Magnitude 7.3485 = area of parallelogram

Scalar projection of a on b3.6467

Vector projection of a on b(1.662, 2.078, 2.494)

Calculation steps

1|a| = √(1.000² + 2.000² + 3.000²) = 3.7417 · |b| = 8.7750
2a·b = (1.000×4.000) + (2.000×5.000) + (3.000×6.000) = 32.0000
3cosθ = a·b/(|a||b|) = 32.0000/(3.7417×8.7750) → θ = 12.93°
4a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁) = (-3.000, 6.000, -3.000)

Special vector quick reference

Vector	Value	Property
i, j, k	(1,0,0) (0,1,0) (0,0,1)	unit basis vectors
0	(0,0,0)	a·0 = 0, a×0 = 0
a ⊥ b	a·b = 0	perpendicular test
a ∥ b	a×b = 0	parallel test (3D)
unit vector of a	a/\|a\|	direction only, \|â\| = 1

Cross product magnitude equals the parallelogram area spanned by a and b — a handy way to compute triangle area from coordinates: S = |a×b|/2.

a·b = 0 ⇔ perpendicular; a×b = 0 ⇔ parallel. Cross product exists only in 3D (and 7D); its direction follows the right-hand rule.

Formulas

Dot: a·b = Σaᵢbᵢ = |a||b|cosθ. Magnitude: |a| = √(Σaᵢ²). Cross (3D): a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁), |a×b| = |a||b|sinθ. Projection of a on b: (a·b/|b|²)b.

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How to Use

Enter values or expressions: type directly into the Vector Calculator — keyboard friendly.
Instant results: everything computes live as you type, no button needed.
Follow the steps: the result panel shows full working and formula substitution.
Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

Vector Calculator is a free online math tool — open and use, no sign-up.
All calculations run locally in your browser; your data never leaves the device.
Key capability: 3D vector add/subtract, dot & cross product, magnitude, angle.
Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

3D vector operations

Add/subtract: a ± b = (a₁±b₁, a₂±b₂, a₃±b₃)

Dot: a·b = a₁b₁+a₂b₂+a₃b₃ = |a||b|cosθ (scalar)

Cross: a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁), perpendicular to both

Magnitude: |a| = √(a₁²+a₂²+a₃²)

Angle: θ = arccos(a·b / |a||b|)

Example: a=(1,2,3), b=(4,5,6): dot = 32, |a|≈3.742, |b|≈8.775, θ ≈ 12.9°.

FAQ

How do I use the Vector Calculator?

Open the page and fill in the input fields — results update automatically. 3D vector add/subtract, dot & cross product, magnitude, angle.

Are the results accurate?

Yes. All calculations use standard mathematical formulas and are verified for accuracy.

Is it free? Do I need to sign up?

Completely free. No registration, login or download required — just open and use.

Does it work on mobile?

Yes. The responsive design works on phones, tablets and desktops.

Is my data saved?

No. All calculations run locally in your browser. Nothing is uploaded or stored.

Related Guides

Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated: 2026-08-07

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Cite this page: YiJianSuan. Vector Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/vector

Results provided by this tool are for reference only and do not constitute professional advice.

Special vector quick reference

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How to Use

Key Points

Formula & How It Works

FAQ

Related Tools

Related Guides

Source & Updates

Special vector quick reference