Unit Vector Calculator

How to Calculate a Unit Vector

A unit vector is a vector with a magnitude (length) of 1 unit. It's used to represent direction without regard to magnitude. Here's a comprehensive guide on how to calculate a unit vector for both 2D and 3D vectors:

Unit Vector Formula

For a 2D vector \(\vec{v} = (v_x, v_y)\), its unit vector \(\hat{u}\) is given by:

\[ \hat{u} = \frac{\vec{v}}{|\vec{v}|} = (\frac{v_x}{|\vec{v}|}, \frac{v_y}{|\vec{v}|}) \]

For a 3D vector \(\vec{v} = (v_x, v_y, v_z)\), its unit vector \(\hat{u}\) is given by:

\[ \hat{u} = \frac{\vec{v}}{|\vec{v}|} = (\frac{v_x}{|\vec{v}|}, \frac{v_y}{|\vec{v}|}, \frac{v_z}{|\vec{v}|}) \]

Where:

• \(\hat{u}\) is the unit vector
• \(\vec{v}\) is the original vector
• \(|\vec{v}|\) is the magnitude of the original vector

Calculation Steps

1. Calculate the magnitude of the vector using the Pythagorean theorem:
   • For 2D: \[ |\vec{v}| = \sqrt{v_x^2 + v_y^2} \]
   • For 3D: \[ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]
2. Divide each component of the vector by its magnitude:
   • For 2D: \[ u_x = \frac{v_x}{|\vec{v}|}, \quad u_y = \frac{v_y}{|\vec{v}|} \]
   • For 3D: \[ u_x = \frac{v_x}{|\vec{v}|}, \quad u_y = \frac{v_y}{|\vec{v}|}, \quad u_z = \frac{v_z}{|\vec{v}|} \]

Example Calculations

2D Example

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

1. Calculate the magnitude: \[ |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
2. Calculate the unit vector components:
   • \(u_x = \frac{3}{5} = 0.6\)
   • \(u_y = \frac{4}{5} = 0.8\)

Therefore, the 2D unit vector is \(\hat{u} = (0.6, 0.8)\).

3D Example

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

1. Calculate the magnitude: \[ |\vec{v}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.071 \]
2. Calculate the unit vector components:
   • \(u_x = \frac{3}{7.071} \approx 0.424\)
   • \(u_y = \frac{4}{7.071} \approx 0.566\)
   • \(u_z = \frac{5}{7.071} \approx 0.707\)

Therefore, the 3D unit vector is \(\hat{u} \approx (0.424, 0.566, 0.707)\).

Visual Representation

This diagram illustrates the original 2D vector \(\vec{v}\) (blue) and its corresponding unit vector \(\hat{u}\) (red). The green dashed circle represents the unit circle, demonstrating that the unit vector has a magnitude of 1.