In electromagnetics, a unit vector is a vector with a magnitude of 1. It's used to express direction without regard to magnitude, making it a fundamental concept in physics and engineering. Unit vectors are particularly useful when dealing with vectors in 2D or 3D spaces, as they help simplify expressions and calculations.
To find a unit vector from a given vector, \(\overrightarrow{v}\), the formula is:
- Divide each component of the vector by the vector's magnitude, \(||\overrightarrow{v}||\).
A unit vector maintains the direction of the original vector but strips away its length, normalizing it:
\[\hat{v} = \frac{\overrightarrow{v}}{||\overrightarrow{v}||}\]
In our exercise example, after deriving the tangent direction vector, the process of normalization gives us the unit vector that is tangent to the circle and points in the desired direction. This process allows us to emphasize direction as opposed to quantity.