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Triangle Inequality Calculator

Online triangle inequality verifier: confirm the magnitude of the vector sum does not exceed the sum of magnitudes.

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|a+b|≤|a|+|b|, equality same direction
Vector a
Vector a
Vector b
Vector b

📖 Tutorial | Triangle Inequality

1. Definition
For any two vectors, |a+b|≤|a|+|b|, with equality when they point the same way; geometrically, two sides of a triangle.
2. Formula & Notation
|a|,|b|Magnitudes
3. Properties & Laws
  • |a+b|≤|a|+|b|;
  • Equality when same direction;
  • ||a|−|b||≤|a+b|;
  • Scalar |x+y|≤|x|+|y|.
4. Steps
  1. Enter two vectors;
  2. Click "Verify";
  3. Or click "Load Example".
5. Worked Examples
Example: a=(3,4), b=(1,2): |a+b|≈7.211≤7.236.
6. Common Pitfalls
Add magnitudes, not components;
Equality needs same direction;
Not an identity.

❓ FAQ | Triangle Inequality

Statement?
|a+b|≤|a|+|b|.
Equality?
Same direction.
Reverse?
Difference bound.
Scalar?
Absolute values.
Geometry?
Two sides.
Three vectors?
Apply twice.
Magnitude?
√(x²+y²).