Vector Magnitude Calculator

How to Calculate Vector Magnitude

The magnitude of a vector is a measure of its length or size. It's a fundamental concept in vector mathematics and has applications in physics, engineering, and computer graphics. Here's a comprehensive guide on how to calculate vector magnitude for both 2D and 3D vectors:

Vector Magnitude Formulas

2D Vector Magnitude

For a 2D vector \(\vec{v} = (x, y)\), its magnitude |\(\vec{v}\)| is given by:

\[ |\vec{v}| = \sqrt{x^2 + y^2} \]

3D Vector Magnitude

For a 3D vector \(\vec{v} = (x, y, z)\), its magnitude |\(\vec{v}\)| is given by:

\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]

Where:

• |\(\vec{v}\)| represents the magnitude of the vector
• x, y, and z are the components of the vector

Calculation Steps

1. Identify the components of the vector (x and y for 2D, x, y, and z for 3D).
2. Square each component.
3. Sum the squared components.
4. Calculate the square root of the sum.

Example Calculations

2D Vector Example

Let's calculate the magnitude of the 2D vector \(\vec{v} = (3, 4)\):

1. Identify components: x = 3, y = 4
2. Square each component: x² = 3² = 9, y² = 4² = 16
3. Sum the squares: 9 + 16 = 25
4. Calculate the square root: \(\sqrt{25} = 5\)

Therefore, the magnitude of \(\vec{v} = (3, 4)\) is 5.

3D Vector Example

Let's calculate the magnitude of the 3D vector \(\vec{v} = (3, 4, 5)\):

1. Identify components: x = 3, y = 4, z = 5
2. Square each component: x² = 3² = 9, y² = 4² = 16, z² = 5² = 25
3. Sum the squares: 9 + 16 + 25 = 50
4. Calculate the square root: \(\sqrt{50} \approx 7.071\)

Therefore, the magnitude of \(\vec{v} = (3, 4, 5)\) is approximately 7.071.

Visual Representation

This diagram illustrates a 2D vector \(\vec{v} = (3, 4)\) and its magnitude. The blue arrow represents the vector, and the red dashed line represents its magnitude (length). For 3D vectors, the concept extends to a third dimension not shown here.