Unit Vectors and Directions

Definition of a Unit Vector

A unit vector is a vector with a magnitude (length) of 1. It indicates direction only and is commonly used to specify directions in space.

• Notation: A unit vector in the direction of vector v is often written as u or ̂v.

• Formula: To find the unit vector in the direction of a given vector v, divide the vector by its magnitude:

• Where is the magnitude (length) of vector .

Finding the Magnitude of a Vector

The magnitude of a vector in two dimensions is given by:

For a three-dimensional vector :

Example: Finding a Unit Vector

Given :

• Find the magnitude:

• Divide each component by the magnitude:

Therefore, the unit vector in the direction of is .

Practice Problems

• Find the unit vector in the direction of .

• Find the unit vector in the direction of .

Solution Steps:

1. Calculate the magnitude .

2. Divide each component of by .

Applications

• Unit vectors are used to describe directions in physics, engineering, and navigation.

• They are essential in expressing vectors in terms of their direction only, independent of magnitude.