Vectors can be defined in multiple ways depending on the context where it is utilized. A vector has both magnitude and direction that is shown over directed line segment where length denotes the magnitude of vector and the arrow indicates the direction from tail to head.

Two vectors are similar if they have same magnitude and direction. The magnitude or direction of a vector with respect to the position doesn’t change. But if you stretch or move the vector from head or tail then both magnitude and direction will change.

In other words, the vector is a quantity having both magnitude and direction. There are scalar quantities that only have magnitude and given a vector measurement. A vector is not important in mathematics only but physics too like aeronautical space, space, traveling guide etc. Pilots use vector quantities while sitting on the plane and taking it to the other direction safely.

Once you are sure on the definition of vector and its usage then next important step is to study the vectors representation. They are represented in the form of a ray and written either in lowercase or upper case. Generally, a single vector is represented in both forms –uppercase and lowercase alphabets. If the vector is written in the form like AB then A is the tail and B is the head.

Vectors are divided into two major categories – one is the dot product, and the other is a cross product. A list of basic formula is available for both the categories to solve the geometrical transformation in 2 dimensions and 3 dimensions. These formulas are frequently used in physics and mathematics. Further, they are widely acceptable for analytical or coordinate geometry problems.

Formula of Magnitude of a Vector

Magnitude of a vector when end point is origin. Let **x **and **y **are the components of the vector,

\[\large \left|v\right|=\sqrt{x^{2}+y^{2}}\]

Magnitude of a vector when starting points are $(x_{1}$, $y_{1})$ end points are $(x_{2}$, $y_{2})$,

\[\large \left|v\right|=\sqrt{\left(x_{2}+x_{1}\right)^{2}+\left(y_{2}+y_{1}\right)^{2}}\]

The formula of resultant vector is given as:

\[\large \overrightarrow{R}=\sqrt{\overrightarrow{x^{2}}+\overrightarrow{y^{2}}}\]

Vector Projection formula is given below:

\[\large proj_{b}\,a=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^{2}}\;\vec{b}\]

The Scalar projection formula defines the length of given vector projection and is given below:

\[\large proj_{b}\,a=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right|}\]

**Unit Vector Formula is given by**

\[\large \widehat{V}=\frac{v}{\left|v\right|}\]

formula of direction is

\[\LARGE \theta =\tan^{-1}\frac{y}{x}\]

Parts in vectors are taken as the angles that are directed towards the coordinate axes. Take an example, if some vector is directed at northwest then its parts would be westward vector and the northward vector. So, vectors are generalized into two parts mostly where names could be different but the concept is same.

With the study of old geometry books, you would know about the evolution of vectors in algebra and how is it beneficial for students. Vectors were initially named as the algebra of segments and directed to displacements. Let us see some of the benefits why students should learn Vectors in school and during higher studies too.

Vectors are important in both physics and mathematics and it was discovered to make the geometry transformations easier. It signifies that quick insights can be gained into Geometry and taken an important part of linear algebra. The popular application of vectors includes – particle mechanics, fluid mechanics, planar description, trajectories calculation, 3D motion etc.

The other area where vectors are used is electromagnetism, analytical geometry, and the coordinate geometry etc. With a clear understanding of Vectors, students not only progress in their career but clear various competitive exams too.