Why Use Position Vectors: Exploring Benefits of Different Coordinates

[Ahmed Elhossiny]

Hello there,
get the notion of position vectors for a particle, but why we use it instead of cartisean coordinates XYZ? What info does the vectors tell us that the cartisean coordinates doesn't tell us?
So if we say a point has coordinates x=2, y=3, z=5

We say its position vector is
r=i 2 + j 3 + k 5

What's the difference! I saw an article saying it's just another way of expressing the position

 

[pixel]

Yes, either form in your example has the same information. But an equation written in vector form is independent of any particular coordinate system. That makes them useful for expressing the laws of physics.

 

[Ahmed Elhossiny]

Yes, either form in your example has the same information. But an equation written in vector form is independent of any particular coordinate system. That makes them useful for expressing the laws of physics.

Doesn't it depend on the X,Y and Z unit vectors? Even if I chose another coordinate system I should express it in terms of the first by (phi)?

 

[pixel]

Doesn't it depend on the X,Y and Z unit vectors? Even if I chose another coordinate system I should express it in terms of the first by (phi)?

You can always transform coordinates to get back to x, y and z. But in some cases, other coordinates are more natural, simpler and useful to apply.

 

[fresh_42]

The Cartesian coordinates of a vector is just one possible of infinitely many possibilities to express a vector according to a basis. But what if you consider the functions? They, too, form vector spaces. Or sequences? The concept of vectors simply restricts the view and methods to what is really needed: addition and stretching (compressing) without the need of one single basis to be used.

 

[Ahmed Elhossiny]

You can always transform coordinates to get back to x, y and z. But in some cases, other coordinates are more natural, simpler and useful to apply.

And that's not applicable in XYZ expression! That's why it's more useful

I always thought it has something to do with direction of movement :D turns out I am wrong

Thanks

 

[Ahmed Elhossiny]

The Cartesian coordinates of a vector is just one possible of infinitely many possibilities to express a vector according to a basis. But what if you consider the functions? They, too, form vector spaces. Or sequences? The concept of vectors simply restricts the view and methods to what is really needed: addition and stretching (compressing) without the need of one single basis to be used.

Can you give me an example of how to express same vector in two cartisean coordinates expression?

 

[fresh_42]

The easiest way is to renumber the coordinates: ##(0,1)## in one coordinate system can be ##(1,0)## in another. Without telling everybody which orientation you use, it is of little help. But you could still say vertical or horizontal unit vector. And why Cartesian coordinates? Sometimes polar coordinates are far easier to handle. And you will have problems to define Cartesian coordinates on the space of all smooth functions on, say a sphere.

 

[Arman777]

Vector's are needed mathematical concepts in physics I think (so in expressing nature).I mean it can't just be some coordinate transformation simplicty.

Think Force or even simpler case, velocity.If velocity would be just scaler, things would be very hard to express.The direction comes naturally these things.And of course we need a magnitude so simply we need a vector.You need vectors to desribe nature correctly.You have a displacement "vector" cause it describes the direction and also magnitude.Lets take a object at point (3,4,5).If you had just this info you can't tell which direction you are, maybe you come from (1,8,9) maybe ( 2,3,4).So a vector can describe the motion of a particle which that's why we use displacement "vector",then from there velocity etc.