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Vectors in Space Calculator

Online 3D vector calculator: enter two vectors to get magnitudes, dot product and the angle; the planar vector extended to three dimensions.

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a·b=x₁x₂+y₁y₂+z₁z₂
Vector a
x
y
z
Vector b
x
y
z

📖 Tutorial | Space Vectors

1. Definition
A 3D vector has magnitude √(x²+y²+z²) and dot product x₁x₂+y₁y₂+z₁z₂, with cosθ=(a·b)/(|a||b|).
2. Formula & Notation
a,b3D vectors
3. Properties & Laws
  • Magnitude √(x²+y²+z²);
  • Dot = sum of products;
  • a⊥b iff dot=0;
  • Same laws as 2D.
4. Steps
  1. Enter the two vectors;
  2. Click "Calculate";
  3. Or click "Load Example".
5. Worked Examples
Example: (1,2,2),(2,1,2): magnitudes 3, dot 8, cos 8/9.
6. Common Pitfalls
Include z;
Sum three squares;
Zero has no angle.

❓ FAQ | Space Vectors

Magnitude?
√(x²+y²+z²).
Dot?
Sum products.
Angle?
arccos.
Perpendicular?
Dot 0.
Vs 2D?
Add z.
cos?
8/9.
Zero?
No angle.