Vector Magnitude Calculator – Guide & Formulas
Calculate the magnitude (length or norm) of any 2D or 3D vector. Get exact and approximate results with step-by-step solutions.
Calculate the magnitude (length, norm, or modulus) of any 2D or 3D vector. The magnitude represents the length of the vector from the origin.
Key Takeaway
Use the free Vector Magnitude Calculator to calculate the magnitude (length or norm) of any 2d or 3d vector. get exact and approximate results with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Vector Magnitude Calculator
- Enter the x, y, and optionally z components of the vector.
- Click "Calculate" to find the magnitude.
- Review the exact and approximate magnitude values.
- The step-by-step solution shows the Pythagorean theorem applied.
The Formula
Variable Definitions
- |v⃗|: Magnitude (length) of the vector
- vx, vy, vz: Components of the vector
- √: Square root operation
Example: Magnitude of Vector (3, 4, 12)
Find the magnitude of the vector v = (3, 4, 12).
- Step 1: Square each component: 3² = 9, 4² = 16, 12² = 144.
- Step 2: Sum the squares: 9 + 16 + 144 = 169.
- Step 3: Take the square root: |v| = √169 = 13.
- Step 4: The magnitude of the vector is exactly 13.
- Step 5: Verification: 3² + 4² + 12² = 9 + 16 + 144 = 169 = 13².
Frequently Asked Questions
What is the magnitude of a vector?
The magnitude (also called length, norm, or modulus) of a vector is its distance from the origin. It is always a non-negative real number.
What is the Euclidean norm?
The Euclidean norm is the most common vector norm, calculated as the square root of the sum of squared components. It corresponds to the straight-line distance.
How is magnitude different from direction?
Magnitude tells you how long the vector is, while direction tells you where it points. A vector has both magnitude and direction.
Can magnitude be negative?
No, magnitude is always non-negative. It is calculated using square roots, which always produce non-negative results.
What is a unit vector?
A unit vector has a magnitude of exactly 1. Any vector can be converted to a unit vector by dividing by its magnitude: v̂ = v⃗/|v⃗|.
What is the magnitude of a zero vector?
The zero vector (0, 0, 0) has a magnitude of 0. It has no direction and represents the absence of a vector quantity.
How do I find magnitude in 2D?
For a 2D vector (vx, vy), the magnitude is |v⃗| = √(vx² + vy²). This is the Pythagorean theorem applied to the vector components.
What are other types of vector norms?
Other norms include the Manhattan norm (sum of absolute values), the max norm (largest absolute component), and the p-norm (sum of |vi|^p)^(1/p).
How is vector magnitude used in physics?
Vector magnitude is used to find the speed from velocity, the force strength, the displacement distance, and many other physical quantities.
What is the relationship between magnitude and dot product?
The magnitude can be found using the dot product: |v⃗| = √(v⃗ · v⃗). The dot product of a vector with itself equals the square of its magnitude.