A unit vector is essentially a directional vector with a standard length.
As defined, unit vectors are vectors whose magnitude is exactly 1 unit. The magnitude of a vector represents its length or size. Therefore, a unit vector is a vector scaled down or up so that its length becomes exactly one unit.
Unit vectors are very useful for different reasons in various fields like physics, engineering, and computer graphics. They are primarily used to indicate direction without adding any information about magnitude.
Key Properties and Uses
- Standard Length: The defining characteristic is that its magnitude (length) is always 1.
- Direction Only: They convey direction alone, making it easy to perform calculations based purely on orientation.
- Forming Other Vectors: Unit vectors serve as building blocks. Specifically, the unit vectors [0,1] and [1,0] can form together any other vector in a two-dimensional space.
- Normalization: Any non-zero vector can be converted into a unit vector by dividing it by its magnitude. This process is called normalization.
Examples in 2D
Consider the standard Cartesian coordinate system. The unit vectors aligned with the axes are fundamental:
- The unit vector along the positive x-axis is often denoted as i or \(\begin{bmatrix} 1 \ 0 \end{bmatrix}\) or [1,0]. Its magnitude is \(\sqrt{1^2 + 0^2} = \sqrt{1} = 1\).
- The unit vector along the positive y-axis is often denoted as j or \(\begin{bmatrix} 0 \ 1 \end{bmatrix}\) or [0,1]. Its magnitude is \(\sqrt{0^2 + 1^2} = \sqrt{1} = 1\).
As the reference notes, the unit vectors [0,1] and [1,0] can form together any other vector. This is because any 2D vector \(\begin{bmatrix} x \ y \end{bmatrix}\) can be written as a linear combination of these unit vectors: \(x \begin{bmatrix} 1 \ 0 \end{bmatrix} + y \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} x \ y \end{bmatrix}\). This illustrates their importance as basis vectors.
In summary, a unit vector is a vector stripped down to its directional essence, having a length of precisely one unit.
