1. ENE 206 Matlab 4 Coordinate systems

2. Vector and scalar quantities Vector scalar A

3. Vectors - Magnitude and direction 1. Cartesian coordinate system (x-, y-, z-)

4. Vector operation in Matlab Cartesian coordinate system Example A = 1a x + 2a y + 3a z >> A = [1 2 3] Find the magnitude of A >> norm(A) or >> abs(A)

5. Scalar product A  B = |A||B|cos  = ABcos  Equivalent definition A  B = A x B x +A y B y +A z B z Scalar projection example_101

6. Cross product A x B = |A||B|sin  = ABsin  Equivalent definition Matlab command is >> cross(A,B)

7. Cross product (cont.) The cross product of the two vectors A = 2a x + 1a y + 0a z and B = 1a x + 2a y + 0a z is shown. The vector product of the two vectors A and B is equal to C = 0a x + 0a y + 3a z. example_102

8. Scalar triple product A  (B x C)=B  (C x A) = C  (A x B) >> dot(A, cross(B,C))

9. Vector triple product A x (B x C)=B(A  C) -C(A  B) >> cross(A, cross(B,C))

10. Volume defined by three vectors originating at a point  v = area of the base x height  v = (|A x B|)(C  a n ) where a n = (A x B)/|A x B| A = [3 0 0]; B = [0 2 0]; C = [0 2 4]; deltav = C  (A x B) example_103

11. Cylindrical coordinate system ( , , z) orthogonal point ( , , z)  = a radial distance (m)  = the angle measured from x axis to the projection of the radial line onto x-y plane z = a distance z (m)

12. Transformation of a vector in cylindrical coordinates to one in Cartesian coordinates A x = A  a x A y = A  a y A z = A  a z where A is in cylindrical coordinates and assumed constant.

13. Dot products of unit vectors in Cartesian and cylindrical coordinate systems cos  -sin  0 sin  cos  0 001

14. Conversion of variables between Cartesian and cylindrical coordinates A conversion from P(x,y,z) to P( ρ, , z) A conversion from P( ρ, , z) to P(x,y,z) Matlab command [ph,rh,z] = cart2pol(x,y,z) Matlab command [x,y,z] = pol2cart(ph,rh,z)

15. The transformation of a vector A = 3a x + 2a y + 4a z in Cartesian coordinates into a vector in cylindrical coordinates. The unit vectors of the two coordinate systems are indicated. figure_112

16. Cylinder creation in Matlab >> [x,y,z] = cylinder(r,n); >> surf (x,y,z) where r = radius n = number of pts along the circumference.

17. Spherical coordinate system ( , ,  ) orthogonal point (r, ,  ) r = a radial distance from the origin to the point (m)  = the angle measured from the positive z-axis (0     )  = an azimuthal angle, measured from x-axis (0    2  ) figure_113

18. Transformation of a vector in spherical coordinates to one in Cartesian coordinates A x = A  a x A y = A  a y A z = A  a z where A is in spherical coordinates and assumed constant.

19. Dot products of unit vectors in Cartesian and spherical coordinates sin  cos  cos  cos  -sin  sin  sin  cos  sin  cos  cos  -sin  0

20. Conversion of variables between Cartesian and spherical coordinate systems A conversion from P(x,y,z) to P(r, ,  ) A conversion from P(r, ,  ) to P(x,y,z) Matlab command [th,phi,r] = cart2sph(x,y,z) Matlab command [x,y,z] = sph2cart(th,phi,r)

21. Convert the Cartesian coordinate point P(3, 5, 9) to its equivalent point in cylindrical and spherical coordinates.