Unit Vector: Definition, Size, Properties

Unit Vector is a vector that represents a unit of measurement.

Vectors are geometric entities that have both a magnitude and a direction of movement. Vectors have two points: a starting point and a terminal point, the latter of which reflects the final position of the point in the equation. Vectors can be used to perform a wide range of arithmetic operations, including addition, subtraction, and multiplication. A unit vector is a vector that has a magnitude of one and has a length of one. For example, the vector v = (1,3) is not a unit vector since its magnitude is not equal to one, i.e., |v| = √(12+32) ≠ 1.; therefore, it is not a unit vector.

When we divide the magnitude of one vector by the magnitude of another vector, every vector can be transformed into a unit vector. A unit vector may also be referred to as a direction vector in some instances. Let us study more about the unit vector, its formula, and a few solved instances to help us understand it better.

Definition of a unit vector: Unit vectors, indicated by the symbol ^a, are vectors with a magnitude equal to one and are hence termed unit vectors. The unit vectors have a length of one. Unit vectors are commonly used to express the direction of a vector in mathematical notation.

Although it has the same direction as an input vector, a unit vector has the same magnitude as an input vector; For a vector called A, a unit vector is  A and A=AA

The Size of a Vector:

When you write down a vector formula, it tells you how big the number is for a given vector. When a vector is made, it has both  direction and magnitude. The individual measures of the vector along the x-axis, y-axis, and z-axis summarises the formula of the magnitude of the vector. When you look at a vector formula, you can see how big it is by looking at how big it is in each of three axes (x,y and z axis). For a  vector A, its magnitude is: |A| As long as you know what direction your vector goes in( along the 3 axis: x axis,y axis and z axis) , you can figure out its size by taking the square root  of the sum of the square of its direction ratios. Let us look at the formula for the magnitude of a vector to make sure we understand what we mean.

To find the magnitude of  a vector that looks like this: A =ai + bj + ck is  

|A|=a2+b2+c2

A unit vector is a vector with magnitude 1

To find a unit vector, u, in the same direction of a vector, v, we divide the vector by its magnitude

        u = vv = 1v v

For a vector v = <a,b> its magnitude is given by

v = a2+b2

The Unit Vector Notation is how you write down the units.

 It looks like a hat or cap: the symbol ‘^’, is called a cap or a hat , for e.g – as ^aa^. Is given by  ^aa^ = a/|aWhere |a| is the magnitude of vector a or norm  or magnitude of the vector a, as shown in the figure. It can be done with a Unit vector formula or with a calculator.

In three-dimensional space, the unit vector is the same thing.

When the unit vectors of ^i, ^j, and ^k are used, they are typically used to represent the unit vectors along the x-axis, y-axis, and z-axis, respectively. In three-dimensional space, any vector that exists may be described as a linear combination of the unit vectors that comprise it. The dot product of two unit vectors is always a scalar quantity, regardless of the units involved. The cross-product of two given unit vectors, on the other hand, produces a third vector that is perpendicular (orthogonal) to both of the first two.

Unit Normal Vector:

In mathematics, a “normal vector” is defined as a vector that is perpendicular to the surface at a specific position. It is also referred to as “normal” to the surface that contains the vector. The unit vector that is obtained after normalising the normal vector is referred to as the unit normal vector, which is also known as the “unit normal”. This is accomplished by dividing a non-zero normal vector by the vector norm.

Formula for Unit Vectors

Because vectors have both a magnitude (Value) and a direction (Direction), they are represented by the arrow ^aa^,, which signifies a unit vector. If we wish to get the unit vector of any vector, we divide the vector’s magnitude by the unit vector of the vector. If you want to represent any vector, you usually use the three-dimensional coordinates x,y,z.

In mathematics, a vector can be represented in one of two ways:

A vector can be shown in two different ways:

Use the brackets to create  = (x, y, z)

In this example,  a =xi + yj + zk is  

There is a formula for how big a vector is:

a= √(x2 + y2 + z2) This is how it looks:

In this example, Unit Vector is Vector/magnitude vectors = Unit Vector.

It is called a unit vector formula, and this is what it looks like

You need to figure out how to find the unit vector

To discover a unit vector that has the same direction as a given vector, divide the vector’s magnitude by the vector’s direction. Consider the case of a vector v = (1,4) with a magnitude of |v|, as an illustration. The unit vector ^v  is obtained by multiplying each component of the vector v by the unit vector |v|, which is oriented in the same orientation as vector v.y, which is its magnitude. If you want to represent any vector, you usually use the three-dimensional coordinates x,y,z.

How do you show a vector in a bracket format?

As you can see, this means that a = a/|a| = (x,y,z)/√(x2 + y2 + z2)  = x/ √(x2 + y2 + z2) , y/√(x2 + y2 + z2) , z/√(x2 + y2 + z2) 

How can I show a vector in a unit vector component format?

It looks like this: a=aa= (xi + yj + zk )/ √(x2 + y2 + z2)  = x/√(x2 + y2 + z2)  .i , y/√(x2 + y2 + z2)  . j, z/√(x2 + y2 + z2) . k

Assuming that the vector is directed, and that x, y, and/or z are the values of the vector along the x-axis (x), the y-axis (y), or the z-axis (z), respectively, a vector is defined as the sum of the vectors’ magnitudes.

a is a unit vector,a is a vector, |a| is the magnitude of the vector, i,j, k are the directed unit vectors along the (x, y, z) axis respectively.

Unit Vector Application

When a vector is expressed as a unit vector, its direction is specified. Both two and three-dimensional planes can include unit vectors. It is possible to represent a vector in terms of its components using its unit vector. A vector’s unit vectors point in the direction of the axes. In three-dimensional space, unit vectors are represented by the formula v = x + y+ z.

Three perpendicular axes will be used to identify the vector v in the 3-D space (x, y, and z-axis). The unit vector on the x-axis is written as  i in mathematical notation. The y-axis unit vector is represented by  j, and the z-axis unit vector is represented by  k

There are so many things that make up the vector, so it can be written like this:

Electromagnetics is about how electric and magnetic forces work together, and how they can be used to make things move. Here, vectors come in very handy to show and do math with these forces. In real life, vectors can show the speed of an aeroplane or a train, where both the speed and the direction of movement are important.

Vector Properties

The properties of vectors help you learn more about vectors and do a lot of calculations with vectors. A few important things about vectors are shown here.

A.B=B.A

AxBB.A

i.i=j.j=k.k=1

i.j=j.k=k.i=0

i×i=j×j=k×k=0

i×j=k; j×k=i; k× i=j

j×i=- k; k×j=-i;i×k=-j

The dot product of two vectors is a scalar that is in the same plane as the two vectors that make up the dot product.

Vectors that make up a cross product are called “perpendicular” vectors, because they are perpendicular to the plane that has them in.