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Vector Magnitude

|v| = √(x²+y²+z²) is the length (Euclidean norm) of vector v. Unit vector v̂ = v/|v| has magnitude 1. 2D: |v| = √(x²+y²). Used for distances and normalization.

Concept Fundamentals
|v| = √(x²+y²)
2D
|v| = √(x²+y²+z²)
3D
v̂ = v/|v|
Unit
|v|² = x²+y²+z²
Squared

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|v| = 0 iff v = 0. Unit vector v̂ has |v̂| = 1. |kv| = |k||v| for scalar k.

Key quantities
|v| = √(x²+y²)
2D
Key relation
|v| = √(x²+y²+z²)
3D
Key relation
v̂ = v/|v|
Unit
Key relation
|v|² = x²+y²+z²
Squared
Key relation

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Why: Magnitude gives vector length, distance from origin, and is used for normalization (unit vectors). Essential in physics (speed, force magnitude) and graphics (normalizing directions).

How: |v| = √(x²+y²) for 2D; add z² for 3D. Unit vector: divide each component by |v|. Squared magnitude |v|² avoids square root when comparing lengths.

|v| = 0 iff v = 0.Unit vector v̂ has |v̂| = 1.
Sources:Khan Academy — Vector MagnitudeWolfram MathWorld — Vector Norm

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🧮 Fascinating Math Facts

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|v| = √(x²+y²+z²).

— Euclidean Norm

̂

Unit vector: v̂ = v/|v|.

— Normalization

Key Takeaways

  • • The magnitude of a vector is its length: ∣vecv∣=sqrtx2+y2+z2|\\vec{v}| = \\sqrt{x^2 + y^2 + z^2}.
  • • For 2D vectors, use ∣vecv∣=sqrtx2+y2|\\vec{v}| = \\sqrt{x^2 + y^2} (set z = 0).
  • • The magnitude is always geq0\\geq 0; it is zero only for the zero vector.
  • • ∣kvecv∣=∣k∣cdot∣vecv∣|k\\vec{v}| = |k| \\cdot |\\vec{v}| for any scalar k.
  • • The unit vector hatv=vecv/∣vecv∣\\hat{v} = \\vec{v}/|\\vec{v}| has magnitude 1.

Did You Know?

Physics

Force, velocity, and acceleration magnitudes are computed using the same formula.

Pythagorean Theorem

The 2D magnitude formula is the Pythagorean theorem: the hypotenuse of a right triangle.

Computer Graphics

Normalizing vectors (making them unit length) is essential for lighting and shading.

Machine Learning

L2 norm (Euclidean norm) is the default for measuring vector length in ML.

Engineering

Resultant force magnitude is found by summing component squares and taking the square root.

Triangle Inequality

|v + w| ≤ |v| + |w| — the sum of lengths is always ≥ the magnitude of the sum.

Understanding Vector Magnitude

The magnitude (or norm) of a vector vecv=(x,y,z)\\vec{v} = (x, y, z) is:

∣v⃗∣=x2+y2+z2|\vec{v}| = \sqrt{x^2 + y^2 + z^2}

For 2D: ∣vecv∣=sqrtx2+y2|\\vec{v}| = \\sqrt{x^2 + y^2}. The unit vector is hatv=vecv/∣vecv∣\\hat{v} = \\vec{v}/|\\vec{v}|.

Expert Tips

Compare Squared Magnitudes

To compare |v| and |w|, compare |v|² and |w|² instead — avoids square roots.

Pythagorean Triples

(3,4), (5,12), (8,15) give integer magnitudes 5, 13, 17.

Zero Vector

For (0,0,0), magnitude is 0. The unit vector is undefined.

Squared Magnitude

|v|² = x² + y² + z² is useful in dot products: v·v = |v|².

Frequently Asked Questions

What is vector magnitude?

The magnitude (or length) of a vector is a scalar: |v| = √(x² + y² + z²).

How do I find magnitude in 2D?

Use |v| = √(x² + y²). Set z = 0 if using the 3D formula.

When is magnitude zero?

Only for the zero vector (0, 0, 0).

What is squared magnitude?

|v|² = x² + y² + z². Useful when you need to compare lengths without square roots.

What is a unit vector?

A vector with magnitude 1. Compute v̂ = v/|v|. Undefined for the zero vector.

Does the formula work for negative components?

Yes. Squaring removes the sign, so (-3, 4) has the same magnitude as (3, -4).

How is this related to the dot product?

v·v = |v|². So |v| = √(v·v).

How to Use This Calculator

  1. Enter vector components (x, y) for 2D or (x, y, z) for 3D.
  2. Click "Calculate" to get magnitude, squared magnitude, and unit vector.
  3. Review the visualization showing the vector and its length.
  4. Copy results for homework or reports.

Disclaimer: This calculator uses standard floating-point arithmetic. For the zero vector, the unit vector is undefined. Results are suitable for educational and professional use.

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