Why do we define our axes to be perpendicular to each other?

Geometry Level 2

What could have happened if we redefined planar coordinate geometry with our axes to be inclined at an angle other than $\frac{\pi}{2}$ ?

This problem is inspired by megh choksi

The axes would have projections on each other and would not be independent. It would not be a consistent idea because the points on the plane would not have unique coordinates. Nothing at all. We just define it to be

\frac{\pi}{2}

to keep things simple. We'd have a new kind of geometry with new sorts of theorems.

2 solutions

Agnishom Chattopadhyay
Dec 13, 2014

Projections are dependent upon the direction you look from. You usually project an inclined vector on the x-axis by dropping its perpendicular because you're interested in 'splitting it into orthogonal components'.

When your axes are inclined at an angle $\theta$ , you'd be not be dropping perpendiculars. You'd be dropping components that touch the other axes at $\theta$ .

Ok, your main question is whether it'd be possible to have axes not perpendicular to each other without breaking things.

The answer is yes, you'd still be able to represent each point uniquely .

Look at the following theorem:

Statement Given any two non-collinear vectors A and B, and another vector C, all on the same plane, there exists unique scalars a and b such that aA + bB = C

Proof Suppose not. Then, we'd have scalars x and y that'd give me xA+ yB = C.

By hypothesis, we have: aA + bB = xA + yB

Or, aA - xA = yB - bB

Or, (a-x)A = (y-b)B

Or, ((a-x)/(y-b))A = B

So, it looks like there is a scalar which when multiplied with the first vector, gives the second vector. That is only possible if the two vectors shared the same direction.

But wait, we defined our vectors to be in different directions. Contradiction!

So, a and b must be unique.

You could think of the coordinates of the point as a position vector, just like C and compare A and B with the axes.

Now, you should be able to see that how the theorem still holds.

Okay, a picture to make it look believable.

Imgur

Is there a good reason to choose $\frac{\pi}{2}$ as the angle between the axes?

There are many. The most important one of them is that it allows us to use the pythagorean theorem to find the distance between two points immediately.

"Nothing at all" does not seem right to me. There will certainly be some changes, like the distance formula will no longer be $\sqrt{ (x_1 -x_2) ^2 + ( y_1 - y_2) ^2 }$ .

Perhaps, clarify what you mean by "what could have happened"?

Calvin Lin Staff - 6 years, 5 months ago

You're correct. Could you help me figuring out what I should have actually put in the option?

Agnishom Chattopadhyay - 6 years, 5 months ago

I would leave it as is for now, till many people report it.

I'm not sure what the best way to express this is. I was tempted to select "we need a new kind of geometry with new theorems", because the distance formula is affected to a slight extent.

Calvin Lin Staff - 6 years, 5 months ago

The choices were a bit confusing.

I'm studying tensor calculus at the moment so affine coordinates pop up quite a bit in examples like creating the notion of the Jacobian matrix. Interesting stuff.

Jake Lai - 6 years, 5 months ago

on respnse to explanation of Agnishom Chattopadhyay .

iam a bit confused. how can you relate "a" and "b" just by the contradicton that both the vectors have opposite direction ?? what is the exact meaning of that "a" and "b" must be unique ?

vipul johri - 6 years, 5 months ago

By stating that they are unique, I mean that there is one and only one possible value of (a,b)

Agnishom Chattopadhyay - 6 years, 5 months ago

Samuel Li
Dec 16, 2014

Suppose the angle was not $\frac{\pi}{2}$ . Rotate the axes so that one of the new axes is concurrent with the x-axis (call this axis "Axis A"). Because the other axis (Axis B) is the only one with a y-component, any given point can be uniquely mapped by correlating its y coordinate with Axis B. The x-component can then be uniquely broken down into Axis A and Axis B. So, every point must have a unique mapping because every point has a unique y-coordinate.

This allows a one-to-one mapping from the new system to the Cartesian coordinate system, and so all fundamentals theorems of geometry hold.

Why do we define our axes to be perpendicular to each other?

This problem is inspired by megh choksi

2 solutions

Imgur

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