Unit Vectors

Unit Vectors

What Are Unit Vectors?

A unit vector is a vector with a magnitude of exactly $1$ . Unit vectors are used to indicate direction without considering magnitude. They are often employed in physics and engineering to simplify calculations and represent directions.

Unit vectors are typically denoted with a “hat” symbol, such as $\hat{i}, \hat{j}, \hat{k}$ .

Standard Unit Vectors in 3D Space

In a Cartesian coordinate system, the standard unit vectors represent directions along the coordinate axes:

$\hat{i}$ : Unit vector along the $x$ -axis ( $[1, 0, 0]$ ).
$\hat{j}$ : Unit vector along the $y$ -axis ( $[0, 1, 0]$ ).
$\hat{k}$ : Unit vector along the $z$ -axis ( $[0, 0, 1]$ ).

These vectors form the basis for representing any vector in 3D space.

Creating a Unit Vector

Any vector $\vec{A}$ can be converted into a unit vector by dividing it by its magnitude. The unit vector $\hat{A}$ is given by:

$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Where:

$\vec{A}$ : Original vector.
$|\vec{A}|$ : Magnitude of $\vec{A}$ , calculated as:
$|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Example Problem

Given a vector $\vec{A} = 3 \hat{i} + 4 \hat{j}$ , find its unit vector.

Solution:

Find the magnitude of $\vec{A}$ :
$|\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$
Divide each component of $\vec{A}$ by its magnitude:
$\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{3 \hat{i} + 4 \hat{j}}{5} = 0.6 \hat{i} + 0.8 \hat{j}$

The unit vector is $\hat{A} = 0.6 \hat{i} + 0.8 \hat{j}$ .

Applications of Unit Vectors

Unit vectors are used in:

Defining directions in physics (e.g., force, velocity).
Expressing components of vectors in coordinate systems.
Solving problems involving motion, fields, and forces.

Key Takeaways

Unit vectors provide a simple way to represent direction in space without magnitude. They are fundamental in vector operations and are widely used in physics and engineering for calculations and problem-solving.