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I have a coordinate system with the Basis $B=(e_x, e_y, e_z)$ and two vectors $r$ and $a$. Now, I want to rotate the basis so that the $e_x$ unit vector points in the direction of the vector $r(x,y,z)$. With the new Basis $B$ I want to get the Vector $a$ with $x$, $y$,$z$

How can I do that?

Thank you! Kate

pink floyd
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Kate
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  • Welcome to MathSE. Please edit your post to include your own attempts at doing this. Seeing your thoughts and what you have already tried makes it much easier for users to post helpful answers. – Mike Pierce Apr 28 '15 at 16:26
  • Can you find the rotation matrix which takes $e_x$ to $r(x,y,z)$? If so, use it on all three of your basis vectors to get your new basis. Then if you want the new coordinates of a vector $a$ whose coordinates you already know in the original basis, use the rotation matrix on this vector as well. –  Apr 28 '15 at 16:26
  • Hi. Thank you for your response. I can calculate the angle between the ex and r, by using the dot product and an rotating axis by using the cross product. But how does the rotating matrix look like? Isn't there a difference if I rotate the vector or the basis? – Kate Apr 28 '15 at 16:32
  • This answer was pretty long, but in it I tried to explain the difference between rotating basis vectors (passive transformation) vs non-basis vectors (active transformation). Also, if you'd like to have someone in particular read a comment (that isn't a comment below their question/ answer), you should include @username so that they are notified. For instance, @Kate –  Apr 29 '15 at 02:14
  • Thank you! What is the differrence between the case http://math.stackexchange.com/questions/180418/calculate-rotation-matrix-to-align-vector-a-to-vector-b-in-3d and my problem, to turn the basis towards the vector? How do the calculation differ? – Kate Apr 29 '15 at 09:16
  • @Kate the question you link to is the same as yours. He wants to find rotation transform that rotates A to B, while you want to rotate r to ex, so it exactly the same thing, but you have different names for your vectors. – vidstige Mar 13 '16 at 06:57

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