Definition and Classification of Vectors

In mathematics and physics, a vector is an element of a vector space that has a specific direction. The vectors have been given specific names in a number of specific vector spaces. An arbitrary starting point is chosen for the sake of convenience, and a Euclidean vector is a geometric object that has both length and direction. It is commonly represented as an arrow. Using vector algebra, such vectors can be added together or scaled in different directions. A vector space is a mathematical term that refers to an ensemble of vectors. It is these objects that are the subject of linear algebra, and they can be distinguished by their dimensions.

At one point in history, vectors were introduced in geometry and physics (typically in mechanics) before the concept of a vector space was formally defined. (In fact, the Latin word vector literally translates as “carrier.”) As a result, when discussing vectors, it is common to refer to them without mentioning the vector space to which they belong. With respect to Euclidean space, spatial vectors (also known as Euclidean vectors) are used to represent quantities that have both magnitude and direction, and they can be added, subtracted, scaled (i.e. multiply by a real number) in order to form a vector space, which is defined as the space between two points on the Euclidean plane.

Magnitude of vectors

The magnitude of a vector can be computed by taking the square root of the sum of the squares of the components of the vector in question. Given a vector A with components (x, y, and z), the magnitude formula for A can be written as follows:

|A| = √(x²+y²+z²)

A vector’s magnitude is represented by a scalar value.

Angle between the two vectors:

It is possible to calculate the angle between two vectors by using the dot product formula. Consider two vectors, a and b, and the angle between them, which is denoted by the symbol θ. In this case, a.b = |a|.|b| cosθ is used to represent the dot product of two vectors. Calculating the angle will be necessary to complete this task. The angle formed by two vectors also serves to indicate the directions of the two vectors. The following formula can be used to determine the value of :

θ = cos^-1[(a.b)/|a|.|b|]

Classification of vectors

The magnitude, direction, and relationship with other vectors are used to categorise the vectors into different types, which are further classified. Let’s take a look at a few different types of vectors and their characteristics:

Zero vectors

Vectors with zero magnitude are referred to as zero vectors, and they are denoted by the symbol 0~ = (0,0,0). The zero vector has no magnitudes and no direction, so it is called a zero vector. It is referred to as the additive identity of vectors in some circles.

Position vector

Three-dimensional space is represented by position vectors, which are used to determine the position and direction of movement of the vectors. In relation to other bodies, it is possible to change the magnitude and direction of position vectors. It is referred to as the location vector in some circles.

Equal vector

When the corresponding components of two or more vectors are equal, two or more vectors are said to be equal. Equal vectors are vectors with the same magnitude and direction as one another. They may have different initial and terminal points, but the magnitude and direction must be the same for both of them.

Parallel vectors

Parallel vectors are defined as two or more vectors that have the same direction but not necessarily the same magnitude when they are in the same plane. Parallel vectors have the same direction, but their angles of direction differ by zero degrees. Antiparallel vectors are vectors whose angles of direction differ by 180 degrees from one another; that is, antiparallel vectors have the opposite directions from one another.

Co-initial vectors

It is referred to as co-initial vectors when two vectors have the same initial point.

Conclusion

A vector is a two-dimensional object that has both a magnitude and a direction associated with it. A vector can be represented geometrically as a directed line segment, with the length of the line segment equal to the magnitude of the vector and an arrow indicating the direction of the vector. In mathematics and physics, a vector is an element of a vector space that has a specific direction. Three-dimensional space is represented by position vectors, which are used to determine the position and direction of movement of the vectors. When the corresponding components of two or more vectors are equal, two or more vectors are said to be equal. Parallel vectors are defined as two or more vectors that have the same direction but not necessarily the same magnitude when they are in the same plane.