8/30/2013 Vector transformation A more rigorous definition of vector starts with the concept that the space is (a) isotopic, so no So far we have used two different ways to describe 8/29/2013 preferred direction, (b) homogeneous, so no 8/29/2013 vector; (a) geometry approach---vector as an arrow, preferred location. A physical quantity such as (b) algebra approach--- vector as components of Rotational matrix displacement or force, should be independent of Rotational matrix Cartesian coordinates. However both approaches the coordinate system we choose. are not very satisfactory and are rather naïve. Let Here we follow the approach of a mathematician and define a vector as a set of three components that transforms in the same manner as a displacement when we change the coordinates. As always, the displacement vector is the model for the behavior of all vectors. 1 2 Rotation about z-axis, using right-hand rule (Counter clockwise). The relationship between , and , can be expressed below: (2) 8/29/2013 8/29/2013 This is rotation about x-axis, again we use right-hand rule, so the y- axis is rotated toward x-axis. The above equations can be combined into one matrix notation; Rotational matrix Rotational matrix For the rotation about y-axis, using right-hand rule, we have: (1) This is a rotation about the z-axis. In the text book, it Now we see that the matrix is a lot different from the previous is a rotation about the x-axis. For rotation about x ones. In order to make the rotation matrix the same as in x and z- axis is given on the next page: axis, we define the rotation about y-axis as clockwise, so the 3 rotational matrix for the y-axis will have similar form as other two 4 axes. For rotation about y-axis, clockwise, the rotation matrix is given Example 1 We want to rotate the z-axis through an angle of 8/29/2013 8/29/2013 φ such that the axis is coincided with line (3) Rotational matrix Rotational matrix Step #1 Now for the most general case, we can rotates the coordinate by Rotates about the z-axis by an angle of an arbitrary angle through an arbitrary direction, and the rotation matrix is given below, however the determination of the matrix elements will be a challenging task, since we only know (4) how to do it through the above three equations (1), (2) and (3) Step #2 Rotates about y’ axis, counterclockwise by an angle of θ (5) 5 6 1 8/30/2013 Re-arrange eq. (5) Problem 1.9 Find the transformation matrix R that describes a rotation by 120° (6) about an axis from the origin through the point (1,1,1). The rotation 8/29/2013 is clockwise. 8/29/2013 z Step #3 Rotational matrix Rotational matrix Rotates about the axis, by an angle of -φ y x (7) By inspection, we can see that the transformation matrix R is given by The final rotation matrix is (9) , , ∙ ∙ 7 8 The rotation on page 8 is equivalent to: 1. Rotates about z-axis for -90°, and then 2. Then rotates about -axis clockwise 90° 8/29/2013 z Rotational matrix y x 0 0 R = 010 0 ° 0 001 φ° 001010 001 010100 100 9 100 001 010 2.