# Why do we choose to have perpendicular axis?

• Avichal
In summary, perperdicular axes are easier to use than angles that are not perperdicular because the inner product is simpler and the coordinate vectors are always perpendicular.
Avichal
Why do we have x-axis perpendicular to y-axis? Why not 45° or something else?
Even if we keep any other angle other than 90° we can still build up all other vectors using them in the plane.

So what is there is 90° that makes it special and simpler?

You can use any angle (and sometimes it is convenient to do so) and have all the same vectors. Several things make 90° special and simpler.

c2=a2+b--2ab cos(t)
This is most simple if cos(t)=cos(90°)=0

very small t are particularly trouble some as two axis are almost the same

to determine the coordinates we must solve
x.i=xii.i+xjj.i
x.j=xii.j+xjj.j

when t=90° this is easy

xi=x.i
xj=x.j

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Avichal said:
Why do we have x-axis perpendicular to y-axis? Why not 45° or something else?
Even if we keep any other angle other than 90° we can still build up all other vectors using them in the plane.

So what is there is 90° that makes it special and simpler?

lurlurf is correct that other axes are possible. In my opinion right angles allow that Pythagorean theorem to be reduced to the sum of squres ot the coordinates. This makes calculation simpler.

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Also the dot product of two perpendicular vectors is 0 so if the x and y axes are perpendicular, and $v_x$ and $v_y$ are unit vectors in the direction of those vectors, then the components of vector v are just $v\cdot v_x$ and $v\cdot v_y$. If the axes were not perpendicular, those formulas would be more complicated.

In more abstract situations, vector spaces of functions, for example, we typically do that the other way around: choose some convenient basis, then define the inner product to make the basis vectors orthogonal.

HallsofIvy said:
Also the dot product of two perpendicular vectors is 0 so if the x and y axes are perpendicular, and $v_x$ and $v_y$ are unit vectors in the direction of those vectors, then the components of vector v are just $v\cdot v_x$ and $v\cdot v_y$. If the axes were not perpendicular, those formulas would be more complicated.

In more abstract situations, vector spaces of functions, for example, we typically do that the other way around: choose some convenient basis, then define the inner product to make the basis vectors orthogonal.

The inner product derives geometrically from orthogonal projection. If the coordinate axes are perpendicular, then orthogonal projection is just picking coordinates.

once you pick one axis, what other natural choice is there for a second axis other than perpendicular? i.e. what is more natural and simpler than angles that are equal?

A related question that I don't understand: Why perpendicular axis are independant of each other?

Avichal said:
A related question that I don't understand: Why perpendicular axis are independant of each other?

It has to do with the inner product of the underlying geometry:

http://en.wikipedia.org/wiki/Dot_product

We might also note that in "differential geometry", we work with surfaces, such as the surface of a sphere, on which we cannot have coordinate curves that are always perpendicular. That causes all sorts of problems, among them that we now have both "covariant" and "contravarient" components of vectors and tensors. If we stick to "Cartesian tensors" in which we only allow "Cartesian coordinate systems" with coordinate curves that are always perpendicular, the distinction between "covariant" and "contravariant" disappears.

I'm not sure since you have posted this in linear and abstract algebra whether my response is relevant or not.

But does this mean

Under what circumstances do we use perpendicular axes and under what circumstances do we use some other axes?

Or do you think we only use perperdicular axes?

The second is far from the truth.
Many different arrangements are in use and the common theme is a blend of ease of presentation and ease of use.

In mathematics you will find cylindrical polar and spherical coordinates.
In cartography, navigation, surveying and fluid mechanics you will find hyperbolic, 'rho-rho' and perhaps even parabolic coordinates.
Look in some engineering texts you will find many graphs with exotic shaped cooordinates.
In geology, soil mechanics and materials science you will find some strange triangular coodinates. These also appear in colour theory in lighting.

Some coordinates have straight line axes, some do not. Look up 'curvilinear coordinates'

go well

## 1. Why do we choose to have perpendicular axis?

The use of perpendicular axis is a common practice in science and engineering, as it allows for simpler and more accurate calculations. This is because perpendicular axis are orthogonal, meaning they intersect at a 90-degree angle, making it easier to measure and calculate certain quantities.

## 2. What are the benefits of using perpendicular axis?

Perpendicular axis offer several benefits, including easier calculations and more accurate results. Additionally, perpendicular axis can simplify the visualization and understanding of complex systems, as they allow for easy separation of forces and components.

## 3. How do perpendicular axis relate to the concept of torque?

Perpendicular axis are crucial in understanding and calculating torque, as they allow us to break down a force into its components and determine its moment arm. This is essential for accurately calculating the torque exerted by a force on an object.

## 4. Can perpendicular axis be used for more than just torque calculations?

Yes, perpendicular axis can be used for a variety of calculations in physics and engineering. They are commonly used in mechanics to analyze systems of forces and determine equilibrium, but they can also be applied in other areas such as optics and electromagnetism.

## 5. What are some real-world applications of using perpendicular axis?

Perpendicular axis are used in many real-world applications, such as designing and building structures, analyzing the stability of objects, and calculating the forces exerted on objects in motion. They are also used in fields such as robotics, aerospace engineering, and biomechanics.

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