Mathematics of Geometry

Many students, incliding me, never talk about geometry in the class of algebra but now I realize that the right way of study of algebra is to connect it with geometry . It is impossible to explain all Algebra with geometry here. So, Today I give you only a way of thinking.One thing that I always quote that Geometry is extremity of every mathematics or every mathematics have their use in Geometry. So, here I give you a list of topics Whose geometry is very Interesting. I think you must have a pen and copy. 

1)Algebra: 

This branch of mathematics have many uses and always have a geometry. If you get a topic in algebra whose geometry is known then its your turn to discover that. We always write many equation like x=0, y=o or x-y=0 ,x^2-y^2=1 etc. 

All these type of equations have definite geometry in Cartesian plan or Euclidean space. Some of above equation are represents a line, some represents curves and and many other eqations are there in Algebra which give you a solid geometry in 3-D.Also we very often use the term "find the root " as:

*what is the root of x^2=1 or x^4=1 ( In real plan and in complex plan). 

*what is the root of the system of equation

                             x-y=1

                            X-2y=2

Try these questions without calculations. ( Hint : Use geometry) 

2) Vector calculus:-

I strictly recommend you that vector calculus is always with geometry becuase this is frequently used in geometry and in engineering, Some examples are as follows: 

 Dot product of two vector:

With the use of  dot product you can find the angle between them since 

                

Dot product has the geometry interpretation as the length of the projection of x onto the unit vector of y when the two vectors are placed so that their tails coincide.

Similarly, there are many terms as cross product, divergence, gradient, curl, etc,. have a geometry Interpretation. 

As we saw two topics whose geometric interpretation is quite interesting. Similary, differential equations are used in geometric problems ,diffrential calculus, trigonometry, integrations, etc have a geometric use. For undergraduate students Groups also have a geometric interpretation and extremity of Groups and rings are called Algebraic geometry. This post may be a trouble for your teachers beacuse now they have to explain more in their classes. But if teacher denies to explain all your question ,never stop yourself and mail me your doubt I will try my best to explain you. 

🙏🙏Thanks🙏🙏

S. K. Ojha