Unit vector — Safekipedia Adventurer

In mathematics, a unit vector is a special kind of vector that has a length of exactly 1. Vectors are like arrows that show direction and size. A unit vector makes it easier to work with just the direction.

Unit vectors are useful in many areas of math and science. They help describe directions clearly, like pointing straight up or to the right. When we need to break down complicated directions, we often use several unit vectors together. This makes calculations simpler and helps us understand how things move in space.

For any vector that isn’t zero, we can create a unit vector that points in the same direction. This is done by dividing the vector by its own length. This process is called normalization, and the result is called the normalized vector. Using unit vectors makes it easier to study directions in normed vector spaces, whether we’re dealing with points on a flat surface or objects moving in three-dimensional space.

Orthogonal coordinates

Cartesian coordinates

Main article: Standard basis

Unit vectors help us show the directions of the x, y, and z axes in 3D space. The main unit vectors for these axes are:

x̂ = [1, 0, 0]
ŷ = [0, 1, 0]
ẑ = [0, 0, 1]

These vectors are important in linear algebra and are often called the standard basis.

Cylindrical coordinates

Spherical coordinates

The unit vectors for spherical coordinates are:

r̂: Points away from the origin.
φ̂: Points in the direction of increasing angle in the x-y plane.
θ̂: Points in the direction of increasing angle from the z-axis.

General unit vectors

Main article: Orthogonal coordinates

Unit vectors are used in many areas of physics and geometry to describe directions in space.

Unit vector	Nomenclature	Diagram
Tangent vector to a curve/flux line	t ^ {\displaystyle \mathbf {\hat {t}} }	A normal vector n ^ {\displaystyle \mathbf {\hat {n}} } to the plane containing and defined by the radial position vector r r ^ {\displaystyle r\mathbf {\hat {r}} } and angular tangential direction of rotation θ θ ^ {\displaystyle \theta {\boldsymbol {\hat {\theta }}}} is necessary so that the vector equations of angular motion hold.
Normal to a surface tangent plane/plane containing radial position component and angular tangential component	n ^ {\displaystyle \mathbf {\hat {n}} } In terms of polar coordinates; n ^ = r ^ × θ ^ {\displaystyle \mathbf {\hat {n}} =\mathbf {\hat {r}} \times {\boldsymbol {\hat {\theta }}}}
Binormal vector to tangent and normal	b ^ = t ^ × n ^ {\displaystyle \mathbf {\hat {b}} =\mathbf {\hat {t}} \times \mathbf {\hat {n}} }
Parallel to some axis/line	e ^ ∥ {\displaystyle \mathbf {\hat {e}} _{\parallel }}	One unit vector e ^ ∥ {\displaystyle \mathbf {\hat {e}} _{\parallel }} aligned parallel to a principal direction (red line), and a perpendicular unit vector e ^ ⊥ {\displaystyle \mathbf {\hat {e}} _{\bot }} is in any radial direction relative to the principal line.
Perpendicular to some axis/line in some radial direction	e ^ ⊥ {\displaystyle \mathbf {\hat {e}} _{\bot }}
Possible angular deviation relative to some axis/line	e ^ ∠ {\displaystyle \mathbf {\hat {e}} _{\angle }}	Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.

Curvilinear coordinates

A coordinate system can use special vectors called unit vectors. These vectors are exactly 1 unit long. They help us describe space. In 3D space, we often use three unit vectors. These vectors are arranged to make calculations easier and to point in the right directions.

Right versor

A unit vector in three-dimensional space was called a right versor by W. R. Hamilton. He used these in his work on quaternions. When the angle is a right angle, the versor is a right versor.

W. R. Hamilton
quaternions
Euler's formula
versor
3-sphere
right angle
imaginary units
complex plane
2-sphere