Unit vector — Safekipedia Discoverer

In mathematics, a unit vector is a special kind of vector that has a length of exactly 1. Vectors are like arrows that show both direction and size, and having one with a length of 1 makes it easier to work with directions alone. Unit vectors are usually written with a little "hat" symbol above the letter, like v̂, which is pronounced "v-hat."

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

For any vector that isn’t zero, we can create a unit vector that points in the exact 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 much easier to study and work with 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 describe the directions of the x, y, and z axes in a 3D space. The standard 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

In general, a coordinate system can be defined using special vectors called unit vectors. These vectors have a length of exactly 1 and help describe the space around us. In ordinary 3D space, we often use three unit vectors, which are arranged in a way that makes calculations easier and ensures they point in the correct directions relative to each other.

Right versor

A unit vector in three-dimensional space was called a right versor by W. R. Hamilton when he developed his quaternions. Every quaternion has a scalar part and a vector part. If the vector part is a unit vector, then its square in quaternions is −1. This helps create versors in the 3-sphere. When the angle is a right angle, the versor is a right versor, with its scalar part being zero and its vector part a unit vector in three-dimensional space.

Right versors extend the idea of imaginary units from the complex plane to range over the 2-sphere in three-dimensional space. A right quaternion is a real multiple of a right versor.

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