Finding perpendicular vector in a skewed coordinate system

In summary, the conversation discusses finding a solution for obtaining V2 (a2, b1, 0) in a skewed coordinate system that is perpendicular to V1 (a1, b1, 0). Suggestions include using an orthonormal coordinate system and taking the inner product of V2 and V1. The conversation also mentions the use of a change of basis matrix and the fact that in either coordinate system, x and y are orthogonal if their inner product is equal to 0.
  • #1
AllenFaust
3
0

Homework Statement


I have an a-b coordinate system which is skewed with an angle = 60 deg. I also have a particle position defined by vector V1 (a1, b1, 0) which follows the coordinate system.

Untitled.png

The problem I have is that I need to get V2 (a2, b1, 0) which is perpendicular to V1.

Homework Equations

The Attempt at a Solution


[/B]
I tried a normal cross product to generate V2

V2 = V1 (a1, b1, 0) x Z (0, 0, 1) = |b1| (a^hat) - |a1| (b^hat) = V2 (b1, -a1, 0)

which works sometimes but not at all points in space. I was thinking this way is only applicable for orthogonal coordinate system.
 
Physics news on Phys.org
  • #2
switch to an orthonormal coordinate system with the appropriate change of basis matrix.
In this new coordinate system, if ## V_1 = P v_1 ##, then you know that ## V_2 = ( B_1, -A_1, 0) ## is orthogonal to ## V_1##, and switch again : ## v_2 = P^{-1} V_2 ##
 
  • Like
Likes AllenFaust
  • #3
geoffrey159 said:
switch to an orthonormal coordinate system with the appropriate change of basis matrix.
In this new coordinate system, if ## V_1 = P v_1 ##, then you know that ## V_2 = ( B_1, -A_1, 0) ## is orthogonal to ## V_1##, and switch again : ## v_2 = P^{-1} V_2 ##

That is actually a good suggestion, however, I am actually thinking of a solution that operates in the skewed coordinate system only. Meaning to say, I can only use the skewed system without the transformation to an orthonormal coordinate system. Thank you.
 
  • #4
Well, one way to do it is take the inner product of V2 and V1 and set that to 0. Note that since you're not in orthogonal coordinates, the product will contain some off diagonal terms. That will give you one equation, you have two unknowns so you'll need one more
 
  • #5
In an orthonormal coordinate system : ##<x,y> = {}^T X Y ##
In a skewed coordinate system ## <x,y > = {}^T X' ({}^T PP ) Y' ##,
where ##X,Y## (resp. ##X',Y'##) are the coordinates of ##x,y## in the orthonormal coordiate system (resp. skewed coordinate system), and ##P## the change of basis matrix from orthonormal to skewed.

No matter which coordinate system you choose, ##x## and ##y## are orthogonal iff ##<x,y> = 0##.
 

What is a perpendicular vector in a skewed coordinate system?

A perpendicular vector in a skewed coordinate system is a vector that is at a 90 degree angle to another vector in the same coordinate system. This means that the two vectors are not parallel and do not lie on the same plane.

How do you find a perpendicular vector in a skewed coordinate system?

To find a perpendicular vector in a skewed coordinate system, you can use the cross product method. This involves taking the cross product of two non-parallel vectors in the coordinate system to get a vector that is perpendicular to both of them.

Why is it important to find perpendicular vectors in a skewed coordinate system?

Finding perpendicular vectors in a skewed coordinate system is important because it allows us to determine the orientation and direction of objects in three-dimensional space. This is crucial for many mathematical and scientific applications, such as calculating forces and velocities in physics.

Can you find a perpendicular vector in any coordinate system?

Yes, you can find a perpendicular vector in any coordinate system as long as you have two non-parallel vectors to use for the cross product method. However, the process may be more complicated in certain coordinate systems, such as polar or spherical coordinates.

Are there any other methods for finding perpendicular vectors in a skewed coordinate system?

Yes, in addition to the cross product method, there is also the dot product method for finding perpendicular vectors in a skewed coordinate system. This method involves taking the dot product of two vectors to determine if they are perpendicular or not. If the dot product is equal to 0, then the vectors are perpendicular.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
20
Views
673
  • Precalculus Mathematics Homework Help
Replies
18
Views
457
  • Precalculus Mathematics Homework Help
Replies
5
Views
456
Replies
7
Views
576
  • Precalculus Mathematics Homework Help
Replies
14
Views
2K
  • Precalculus Mathematics Homework Help
Replies
6
Views
4K
  • Precalculus Mathematics Homework Help
Replies
3
Views
5K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Precalculus Mathematics Homework Help
Replies
20
Views
2K
  • Precalculus Mathematics Homework Help
Replies
10
Views
2K
Back
Top