Question

Another coordinate with respect to the basis question.. ? Suppose that a rectangular x'y' coordinate system is obtained by rotating a rectangular xy coordinate system about the origin through the angle theta = 3pi/4. Find the rotations equations that express x'y'-coordinates in terms of xy coordinates and use those equations to find the x'y' coordinates of the point whose xy coordinates are (-2,6).

Answer 1

I don't understand this now that it integrates theta and angles into the questions.. Can someone explain

Answer 2

U want rotation matrix?

Answer 3

Is that what I'm looking for? I thought it was a rotation equation

Answer 4

Well it is in a way, just in matrix form..

Answer 5

If u use ((cos theta, -sin theta)(sin theta, cos theta)) is a rotation through theta radians about the origin.

Answer 6

I'm kind of confused b/c we're given the coordinates is it like before where we'd need vectors and a basis?

Answer 7

like B={v_1, v_2,v_3} and W or something like that?

Answer 8

Nothing stopping u from treating a point as a vector here.

Your "basis" is the xy frame.

If u apply a matrix ((a,b)(c,d)) to (x,y) u get

(ax+by,cx+dy) Ç= x(ac) + y(bd)

Answer 9

ignore the Ç,sorry...

Answer 10

Rotating a vector and rotating the plane are "equivalent" in some sense.

Answer 11

\[x'= xcos \theta-ysin \theta\]
\[y'= xsin \theta+ycos \theta\]

Answer 12

Yes, that's the matrix applied...

Answer 13

blah still kind of lost, where'd you guys get these equations from

Answer 14

ok all possible pairs of input can however not exist

Answer 15

are those used for rotation around a vector

Answer 16

No, that's a different thing....

Answer 17

U understand that the equations phi posted are the ones u get from the rotation matrix?

Answer 18

Yes I do those are rotation equations for the plane?

Answer 19

If I rotate the plane then a vector will get carried along with it....

Answer 20

http://en.wikipedia.org/wiki/Rotation_matrix
Lots of boring stuff here!

Answer 21

Or a point if u prefer (I hate that they do double duty but...).

Answer 22

Oh yeah that make sense about the plane and vector thing but so if I put in the theta into the equation I get the equation right but what do I do with the coordinates

Answer 23

Just put your x,y into phi's equations plus your angle and u get x',y'

Answer 24

The coordinates (x,y) are used in the rotation eq to find (x',y')

Answer 25

What estudier said.

Answer 26

okay what do those values of coordinates represent again

Answer 27

Which values?

Answer 28

the x' and y' is that the new coordinates after the system is rotated?

Answer 29

Yup

Answer 30

Oh okay I kind of follow where you guys are going at. Thanks a lot of all your help!! Very much appreciated

Answer 31

ur welcome.