How to Calculate the Magnitude of a Vector

Vector Magnitude Calculator

X Component (i):

Y Component (j):

Z Component (k) (Optional for 3D):

Understanding Vector Magnitude

In physics and mathematics, a vector is a quantity that has both magnitude and direction. Unlike a scalar quantity (like temperature or mass) which only has magnitude, a vector describes movement or force in a specific direction. Common examples of vector quantities include velocity, acceleration, and force.

What is Vector Magnitude?

The magnitude of a vector represents its "length" or "strength" without considering its direction. It tells you how large the vector quantity is. For instance, if a vector represents velocity, its magnitude is the speed. If it represents force, its magnitude is the amount of force applied.

The magnitude of a vector is always a non-negative scalar value. It is often denoted by double vertical bars around the vector symbol, like ||v|| or simply |v|.

The Formula for Vector Magnitude

The calculation of a vector's magnitude depends on its dimensions (2D or 3D). The principle, however, remains the same: it's derived from the Pythagorean theorem.

For a 2D Vector

If a vector v is represented in a 2D Cartesian coordinate system as (x, y), its magnitude is calculated using the formula:

|v| = √(x² + y²)

Where:

x is the component of the vector along the X-axis.
y is the component of the vector along the Y-axis.

For a 3D Vector

If a vector v is represented in a 3D Cartesian coordinate system as (x, y, z), its magnitude is calculated using the formula:

|v| = √(x² + y² + z²)

Where:

x is the component of the vector along the X-axis.
y is the component of the vector along the Y-axis.
z is the component of the vector along the Z-axis.

How to Use the Calculator

Our Vector Magnitude Calculator simplifies this process for you:

X Component (i): Enter the value of the vector's component along the X-axis.
Y Component (j): Enter the value of the vector's component along the Y-axis.
Z Component (k) (Optional for 3D): If your vector is 3-dimensional, enter its component along the Z-axis. If it's a 2D vector, you can leave this field blank or enter '0'.
Click the "Calculate Magnitude" button.

The calculator will instantly display the magnitude of your vector.

Examples

Example 1: 2D Vector

Let's say you have a 2D vector v = (3, 4).

X Component (i) = 3
Y Component (j) = 4
Z Component (k) = 0 (or left blank)

Using the formula: |v| = √(3² + 4²) = √(9 + 16) = √25 = 5.

The magnitude of the vector (3, 4) is 5.

Example 2: 3D Vector

Consider a 3D vector u = (1, 2, 2).

X Component (i) = 1
Y Component (j) = 2
Z Component (k) = 2

Using the formula: |u| = √(1² + 2² + 2²) = √(1 + 4 + 4) = √9 = 3.

The magnitude of the vector (1, 2, 2) is 3.

Example 3: Vector with Negative Components

Let's calculate the magnitude of vector w = (-5, 12).

X Component (i) = -5
Y Component (j) = 12
Z Component (k) = 0 (or left blank)

Using the formula: |w| = √((-5)² + 12²) = √(25 + 144) = √169 = 13.

The magnitude of the vector (-5, 12) is 13.