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 July 30th, 2013, 08:18 AM #1 Newbie   Joined: Jul 2013 Posts: 4 Thanks: 0 Rotation of a Cartesian Coordinate System How can I rotate the Cartesian coordinate system ( i=(1,0,0); j=(0,1,0); k=(0,0,1) ) so that angles between new and the old axes be equal to a, b and c, respectively? Here a is angle between i and i' etc. Is there any connection to the Euler angles transformation?
 July 30th, 2013, 03:23 PM #2 Global Moderator   Joined: Dec 2006 Posts: 19,967 Thanks: 1850 Do you realize that the angles can't be chosen arbitrarily?
July 30th, 2013, 04:21 PM   #3
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Quote:
 Originally Posted by skipjack Do you realize that the angles can't be chosen arbitrarily?
No, I did not. Why can't be angles chosen arbitrarily? You mean that there will not be a unique solution?

 July 30th, 2013, 05:23 PM #4 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Rotation of a Cartesian Coordinate System What angles, exactly, are you talking about? Given a single point, in a three dimensional Cartesian coordinate system, we can define the three angles between the straight line from the origin, (0, 0, 0), to the point and the three (positive) coordinate axes. If we call those angle a, b, and c. However, it can be shown that the three "direction cosines", cos(a), cos(b), cos(c), together, give the unit vector $cos(a)\vec{i}+ cos(b)\vec{j}+ cos(c)\vec{k}$ in the direction of that point. That is, we must have $cos^2(a)+ cos^2(b)+ cos^2(c)= 1$.
July 30th, 2013, 05:42 PM   #5
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Re: Rotation of a Cartesian Coordinate System

Quote:
 Originally Posted by HallsofIvy What angles, exactly, are you talking about? Given a single point, in a three dimensional Cartesian coordinate system, we can define the three angles between the straight line from the origin, (0, 0, 0), to the point and the three (positive) coordinate axes. If we call those angle a, b, and c. However, it can be shown that the three "direction cosines", cos(a), cos(b), cos(c), together, give the unit vector $cos(a)\vec{i}+ cos(b)\vec{j}+ cos(c)\vec{k}$ in the direction of that point. That is, we must have $cos^2(a)+ cos^2(b)+ cos^2(c)= 1$.
I will try to restate the problem. I am looking for a new basis vectors: i', j' and k' for 3D space which are in angles ?, ? and ? relative to the old basis vectors: i, j and k.
So, ? is angle between i and i', ? is angle between j and j' and ? is angle between k and k'.

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