Equality of vector

Mainly there are two types of quantities. Anything is denoted with the help of quantities. These quantities are termed scalar and vector. Scalar means the quantity which only has a magnitude that is an integer, decimal, or fractional value. But in the case of a vector, a quantity that has magnitude and direction is known as a vector quantity. 

With the help of an example, we could know what exactly this vector means. For instance, if we talk about mass, time, distance, speed, etc., they are considered scalar quantities. And terms such as displacement, force, velocity, acceleration, etc., are allotted in the section of a vector quantity. 

Furthermore, vectors are subdivided into some categories. And it has specific rules and regulations of itself. With the help of those rules, we could define its properties and which type of vector it is. Vectors are very useful in mechanical life and in our day-to-day life too. There are various applications of vectors around us, and it makes our life simple to some extent.

A brief note on Vectors

In physics, a vector is a quantity with both magnitude and direction. It’s usually represented by an arrow with the same direction as the amount and a length proportional to the amount of the quantity. Even though it has magnitude and direction, a vector has not had a position. A vector is not altered if it is shifted parallel to itself as long as its length is not modified. To qualify as a vector, a quantity with magnitude and direction must also follow specific combination criteria. Vector addition, expressed symbolically as A +B+ C, is one. The vectors are majorly written in uppercase letters. 

The Latin word vector means “carrier” and must “transport” point A to point B. Astronomers studying the planetary revolution around the Sun in the 18th century were the first to use it. The distance between the two places is the magnitude of the vector, and the direction is the path of displacement from A to B. Many arithmetic operations on real numbers, such as addition, subtraction, multiplication, and negation, have vector analogues that fulfil commutativity, associativity, and distributivity standard algebraic laws. Euclidean vectors are examples of the more generic concept of vectors defined simply as members of a vector space, as defined by these operations and laws. Vectors are significant in physics because they can explain the velocity and acceleration of a moving object and the forces acting on it. Many additional physical quantities can be thought of in the same way. Although most of them don’t represent distances (save for location or displacement), the length and orientation of an arrow can nevertheless be used to communicate their direction and magnitude. The coordinate system used to define a physical vector impacts its mathematical representation. Pseudovectors and tensors are other vector-like objects that express physical quantities and transform similarly when the coordinate system changes.

Types of vectors

There are numerous types in which we can find and describe the vector. Each of them are unique, though easy to understand. They have certain properties to describe them. 

•  Co-initial Vector

 Co-initial vectors are a form of vector in which the beginning points of two or more distinct vectors are the same. All vectors in this sort of vector begin from the same point. The vectors are called co-initial vectors because their origin points are the same. For example, when we have two AB and AC vectors, they are referred to as co-initial vectors because they share the same initial point, A.

•  Collinear Vectors

The collinear vector is another sort of vector in which two or more vectors, regardless of magnitude or direction, are parallel. Because they are parallel, they never cross paths. In nature, both vectors have the same direction. Collinear vectors, for example, are those in which vector an is in the x-direction, and vector b is also in that direction. Both vectors have the exact dimensions in nature. The cross product from both collinear vectors always seems to be equal to zero, another property of collinear vectors. Paralleled vectors are another name for collinear vectors.

•  Zero Vectors

The zero vector is another sort of vector in which the value of the vector is zero, and also the origin and endpoint points of the vector are the same. For example, if the magnitude of vector A and B are the same but direction are opposite to each other, then the vector  A +B is described as a zero vector. The magnitude of the zero vector is always 0, and its direction is undetermined. The zero vector has no constituents and will not point in any direction.

•  Unit Vector

The unit vector is a subtype of a vector with a value based on the length of one unit. Unit vectors are defined as any vectors with a magnitude of one. If there is a vector x  with the magnitude x, then the unit vector is x, which has the same direction as vector x and the same magnitude. Even if the two vectors had the same value, they are not deemed equal unless they simultaneously have the same direction.

•  Position Vector

A position vector is a vector in which the origin point is set to O, and one random point in the space is designated as A. The position vector with the reference origin O is then known as vector AO. In three-dimensional Cartesian geometry, the position vector is primarily used to indicate the placement or position of a point. And any reference origin can be used to determine the position.

•  Coplanar Vectors

Coplanar vectors have three or more vectors in the same plane, or that can lie in the parallel plane. There is always the chance of finding any two different vectors in the same plane, referred to as coplanar vectors. The vector triple product of the three vectors is always equal to zero, which is another property of coplanar vectors. Based on linear vectors are always coplanar vectors.

•  Unlike and Like Vectors 

Like vectors are the type of vectors that have the same direction and are referred to as such. Unlike vectors are vectors that have the same direction but are in opposite directions. 

•  Equal Vectors

Equal vectors are the sort of vector in which two or more vectors with the same magnitude and direction are considered equal.

•  Displacement Vector

The displacement vector is the sort of vector that occurs when one vector is shifted from its original position. For example, suppose an object is present at position A at time =0 and then moves to point B at time =t after some time. The vector distance between the object’s starting and final points can determine the displacement.

10 Negative Vector

A negative vector is a form of vector in which the value of both vectors is equal, but the direction of both vectors is opposite. Negative vectors are the name for this sort of vector. Consider the case where we have two individual vectors, a and b. Then we can write them as

 a = -b. This is known as a negative vector.  

The magnitude and direction of a vector can be separated into two parts. The horizontal and vertical components may be obtained by constructing a right triangle with the vector to be studied as the hypotenuse. The horizontal component of the triangle is the bottom edge, while the vertical component is the side opposite the angle.

The length of the two components can be calculated using the angle that the vector makes with the horizontal. The horizontal and vertical components may be obtained by constructing a right triangle with the vector to be studied as the hypotenuse. The horizontal component of the triangle is the bottom edge, while the vertical component is the side opposite the angle.

The length of the two components can be calculated using the angle that the vector makes with the horizontal.