Measure complex vector magnitude with reliable formulas and quick validation. Adjust dimensions easily for practice. Export clear results, tables, and printable summaries for study.
| Component | Complex Value | Modulus | Modulus Squared |
|---|---|---|---|
| z1 | 3 + 4i | 5 | 25 |
| z2 | 1 - 2i | 2.2361 | 5 |
| z3 | -2 + 2i | 2.8284 | 8 |
| Euclidean Length | √38 ≈ 6.1644 | ||
For each complex component, let z = a + bi. Its modulus is |z| = √(a² + b²).
The standard complex vector length is the Euclidean norm:
||v||₂ = √(|z₁|² + |z₂|² + ... + |zₙ|²)
The squared Euclidean norm is:
||v||₂² = |z₁|² + |z₂|² + ... + |zₙ|²
The 1-norm is:
||v||₁ = |z₁| + |z₂| + ... + |zₙ|
The infinity norm is:
||v||∞ = max(|z₁|, |z₂|, ... , |zₙ|)
The p-norm is:
||v||ₚ = (|z₁|ᵖ + |z₂|ᵖ + ... + |zₙ|ᵖ)^(1/p)
If the selected norm is nonzero, the normalized component is:
uᵢ = zᵢ / ||v||
The Euclidean version also matches the Hermitian form √(v* · v).
Complex vectors appear in algebra, engineering, and signal work. Their length measures overall size. That size helps compare states, signals, and solutions. A strong calculator saves time. It also reduces manual mistakes during repeated practice or technical review.
This page computes the modulus of every complex component first. Each entry uses a real part and an imaginary part. Those moduli then build the full vector norm. The main result can follow the Euclidean norm, the 1-norm, the infinity norm, or a general p-norm.
A complex component has the form a + bi. Its magnitude is based on the Pythagorean relation. After finding each component magnitude, the calculator combines them into a vector length. This structure is useful in linear algebra, quantum mechanics, optimization, and numerical methods.
The squared Euclidean norm avoids an extra square root. That makes algebra cleaner. It is common in proofs, gradient methods, and energy expressions. Many textbooks write vector energy in that form because it is compact and easy to differentiate.
This calculator also returns component angles and normalized components. Angles describe direction in the complex plane. Normalized components scale the vector to unit size when possible. That helps when you need fair comparisons between vectors of different lengths.
Students can check homework quickly. Teachers can build examples. Engineers can inspect signal vectors. Analysts can test custom p-norm settings. Because the layout is simple, the tool works well for short checks and detailed reviews.
A complex vector length measures the size of a vector whose entries contain real and imaginary parts. The standard version is the Euclidean norm, built from each component modulus.
For a component a + bi, the modulus is √(a² + b²). The calculator applies this to every component before combining them into the selected vector norm.
The 1-norm adds component moduli directly. The 2-norm squares them, sums them, then takes the square root. The 2-norm is the standard vector length in most math courses.
Use the infinity norm when you need the largest component magnitude only. It is useful in error bounds, stability checks, and max-size comparisons.
Normalization divides each component by the selected vector norm. The result is a unit vector when the original length is not zero. Zero vectors cannot be normalized.
Each angle shows the direction of a complex number in the plane. This helps you inspect phase behavior and understand how the vector is distributed across components.
Yes. The inputs accept decimals, negative numbers, and zero values for both real and imaginary parts. Blank fields are treated as zero during calculation.
CSV is useful for spreadsheets, logging, and later analysis. PDF is useful for reports, printing, and sharing clean result summaries with classmates or colleagues.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.