Unit Vector:

A unit vector is a vector that has a magnitude of 1. In other words, its length or size is equal to 1. Unit vectors are commonly used in mathematics and physics to represent directions or orientations. They provide a convenient way to describe the direction of a vector without worrying about its magnitude.

Properties of a Unit Vector:

1. Magnitude: A unit vector has a magnitude of 1. This means that its length or size is equal to 1. It can be represented as |u| = 1, where u is the unit vector.

2. Direction: A unit vector represents the direction of another vector. It points in the same direction but does not have the same magnitude. It is often denoted by a letter with a hat (^) symbol, such as ȳ.

3. Normalization: To convert a vector into a unit vector, we perform a process called normalization. This involves dividing each component of the vector by its magnitude. The resulting vector will have a magnitude of 1 and represent the same direction as the original vector.

Example:

Consider a vector u = (3, 4). To find its unit vector, we need to divide each component by the magnitude of the vector.

Magnitude of u = √(3^2 + 4^2) = √(9 + 16) = √25 = 5

To normalize u, we divide each component by 5:

ȳ = (3/5, 4/5) = (0.6, 0.8)

Therefore, the unit vector of u is ȳ = (0.6, 0.8).

Importance:

Unit vectors are important because they allow us to describe the direction of a vector without worrying about its magnitude. They are used in various applications, such as calculating forces, representing directions in 3D space, and solving mathematical equations involving vectors. Unit vectors also play a crucial role in linear algebra, where they form a basis for vector spaces.

Conclusion:

A unit vector is a vector with a magnitude of 1. It represents the direction of another vector without the consideration of its magnitude. Unit vectors are commonly used in mathematics and physics to describe orientations, calculate forces, and solve mathematical equations involving vectors.