Unit Vectors

Vector units have both magnitude and direction. However, there are times when only the magnitude is considered and not the direction. In such cases, vectors are frequently regarded as unit length. These unit vectors are commonly used to represent direction, with the magnitude provided by a scalar coefficient. A vector decomposition is the sum of a unit vector and scalar coefficients.

Vectors are a type of geometric entity with magnitude and direction. Vectors have a starting point and a terminal point that represents the point’s final position. Vectors can be subjected to a variety of arithmetic operations, including addition, subtraction and multiplication. The unit vector has a value equal to one.

Vector v = (1,3), for example, is not a unit vector because its magnitude is not equal to 1. i.e.,

 |v| = √(12+32) ≠ 1.

When we divide the magnitude of one vector by the magnitude of another, we get a unit vector. A direction is another name for a unit vector.

Component form of vector

Any vector p in the Cartesian coordinate system can be represented by its unit vectors. The unit vectors in the directions of the x, y, and z-axes are given by i,  j, and k, respectively. The vector’s position in space with respect to the origin of the given coordinate system can be represented as follows:

p=xi+yj+zk

Vector properties

The properties of vectors are useful for gaining a thorough understanding of vectors as well as performing a variety of vector-related calculations. A few important vector properties are listed here.

• A.B=B.A
• AхB≠B х A
• i.j=j .k=k.i=o
• i ✖ i=j ✖ j=k ✖ k=0

Unit vector symbol

Unit vectors, denoted by a, are vectors with magnitude equal to one. Unit vectors have a length of one. Unit vectors are commonly denoted by a.

Three-dimensional unit vector

The unit vectors  are the vectors along the x, y, and z  axis, respectively. Each vector in three dimensions can be expressed as a linear combination of these unit vectors. The sum of two unit vectors’ dot products is always a scalar quantity. The cross-product of two given unit vectors, on the other hand, yields a third vector that is perpendicular (orthogonal) to both of them.

A vector’s magnitude

The vector formula returns the numeric value of a given vector. The magnitude of a vector formula is the sum of the vector’s individual measures along the x, y, and z axes. The magnitude of a vectorA is represented by the symbol |A|. The magnitude of a vector with directions along the x, y, and z axes is the square root of the sum of the squares of its direction ratios.

The magnitude of a vector A = ai+ bj+ ck is:

|A|= √(a2+b2+c2)

Unit vector formula

A vector has magnitude and direction. A unit vector has a magnitude of one. It’s conjointly mentioned as a direction vector.

Two 2D direction vectors are d1 and d2. 2D spatial directions are represented in a way that is equivalent to points on the unit circle.

The unit vector formula is provided by:

The first step is to express the coordinates in brackets: v = (x,y,z).

As an alternative, three-unit vectors are used, one for each axis, and the vector is represented by the following notation: v=xi+yj+zk. Furthermore, the vector has the following magnitude:

|v| = x2+y2+z2

Unit vector = vectormagnitude

If we write it in bracket format then

v= v|v|= (x,y,z)x2 +y2+z2= (x)x2 +y2+z2 ,(y)x2 +y2+z2 ,(z)x2 +y2+z2

Unit tangent vector 

Assuming v(t) is a smooth vector-valued function with a zero-threshold, any vector that is parallel to V'(t) is considered to be tangent to the graph of v(t) at the zeroth degree. It is most commonly used to represent the direction V’ (t) rather than the magnitude of this same direction.

We are thus examining the unit vector pointing in V ‘(t) direction. In this case, the following is the unit tangent vector definition:

Examine the smooth function V’ (t) on the open interval I. Thus, this example denotes the unit tangent vector as t(t).

t(t) =V'(t)/|V'(t)|

Unit normal vector 

For simplicity, consider the vector-valued function V (t), in which the unit tangent vector, denoted by the symbol t'(t), is smooth on the open interval. N (t) is defined as the unit normal vector n (t).

n(t) =t'(t)/|t'(t)|

The primary unit vector is another name for this vector.

Applications of unit vector

• In Physics, a unit vector is a vector that represents one unit.
• It is a vector with an equal magnitude and a certain direction known as the unit vector. The sole direction is determined by a unit vector, the smallest vector. Dimensions and units are not given.
• The x-axis, y-axis, and z-axis are all represented in a rectangular coordinate system.
• They are parallel and perpendicular to the other unit vectors in different coordinate axes.

Conclusion

The term direction vector refers to a unit vector that is used to represent spatial direction and is commonly abbreviated as d. Two-dimensional spatial directions are equivalent to points on the unit circle, whereas three-dimensional spatial directions are equivalent to a point on the unit sphere.

Unit vector = vectormagnitude