Vectors and Polar Coordinates

1. Vectors and Polar Coordinates Lecture 2 (04 Nov 2006) Enrichment Programme for Physics Talent 2006/07 Module I

3. 2.1 Vectors and scalars • Vector: quantity having both magnitude and direction, e.g., displacement, velocity, force, acceleration, … • Scalar: quantity having magnitude only, e.g., mass, length, time, temperature, …

4. 2.1 Vectors and scalars Fundamental definitions: • Two vectors and are equal if they have the same magnitude and direction regardless of the initial points • Having direction opposite to but having the same magnitude

5. 2.1 Vectors and scalars • Addition:  • subtraction: 

6. 2.1 Vectors and scalars Laws of vector

7. 2.1 Vectors and scalars • Null vector: vector with magnitude zero • Unit vector: vector with unit magnitude, i.e., . • Rectangular unit vectors , and . unit vector , (x, y, z) are different components of the vector . • Magnitude of :

8. 2.1 Vectors and scalars Example: Find the magnitude and the unit vector of a vector Write: , where Magnitude: Unit vector:

9. 2.1 Vectors and scalars Dot and cross product • Dot product: , where q is the angle between vectors and . Laws of dot product: q

10. 1. 2. 2.1 Vectors and scalars Example: Evaluate the dot product of vectors 1. and 2. and

11. q q 2.1 Vectors and scalars Dot and cross product • cross product: , where q is the angle between vectors and . is a unit vector such that , andform a right-handed system. area of the parallelogram

12. 2.1 Vectors and scalars Dot and cross product • Laws of cross product:

13. 1. 2. 2.1 Vectors and scalars Example: Evaluate the cross product of vectors 1. and 2. and

14. (v1, v2) 2.2 Matrix operations of rotations Vectors in 2-dimensions • a vector in a 2-dimensional plane can be written as , • and are called the basis vector, since any vector can be written as a linear combination of the basis vector

15. (v1, v2) (v1’, v2 ’) base vectors are not unique! 2.2 Matrix operations of rotations Vectors in 2-dimensions • any vector in R2 can be written as • and are called the base vectors, since any vector can be written as a linear combination of the base vectors, namely Is base vectors unique?

16. 2.2 Matrix operations of rotations Vectors in 2-dimensions • Generally, let and are base vectors, i.e. • Base vectors are said to be orthonormal if • Hence, and are example of orthonormal base vectors.

17. (v1, v2) (v1’, v2 ’) 2.2 Matrix operations of rotations Vectors in 2-dimensions • Let both and are orthonormal base vectors, i.e., • using different coordinate system to represent is possible. • since How to express them in matrix form?

18. (v1, v2) (v1’, v2 ’) 2.2 Matrix operations of rotations Vectors in 2-dimensions or in matrix form:  Note are orthogonal.

19. 2.2 Matrix operations of rotations Vectors in 2-dimensions Hence, an orthogonal matrix R acts as transformation to transforms a vector from one coordinates to another, i.e.,

20. 2.3 Polar coordinates The position of the “Red Point” can be represented by (r, ) instead of (x, y) in Cartesian Coordinates. y r = magnitude of the position vector  = angle of the position vector and the x-axis x O

21. 2.3 Polar coordinates In Polar Coordinates, we define two new base vectors instead of in Cartesian Coordinates. y : a unit vector in the direction of increasing r (i.e. -direction) : a unit vector in the direction of increasing  x O

22. 2.3 Polar coordinates Any vector on the 2D plane can be expressed in terms of and : y In particular, the position vector is given by x O

23. 2.3 Polar coordinates Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y): Cylindrical Coordinates: Cartesian Coordinates:

24. 2.3 Polar coordinates Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y): :

25. 2.3 Polar coordinates Conversions between Polar Coordinates (r, ) and Cartesian Coordinates (x, y): :

26. 2.3 Polar coordinates Differentiating a vector in Polar Coordinates (r, ): :

27. 2.3 Polar coordinates Central Force Field Problem: External Torque = 0:  Conservation of Angular Momentum

28. Box 2.1Angular momentum • Recall: momentum , where m is the mass, is a measure of the linear motion of an object. • The angular momentum of an object is defined as: a measure of the rotational motion of an object. a

29. Box 2.1Angular momentum • As linear momentum, an object keeps its motion unless an external force is acted; • An object has a tendency to keep rotating unless external torque is acted. It is the conservation of angular momentum. The conservation of angular momentum explains why the Earth always rotates once every 24 hours.

30. 2.3 Polar coordinates • Area swept out in a very small time interval:

31. 2.3 Polar coordinates • In general, planets’ orbits are elliptical • To describe its motion,

32. 2.3 Polar coordinates • is constant if angular momentum is conserved and m is unchanged.

34. 2.3 Polar coordinates Johannes Kepler (開普勒) 1571 - 1630 • This is in fact one of his famous three laws of planetary motion, which are deduced from Tycho’s 20 years observation data.

35. 2.3 Polar coordinates The second law of planetary motion: equal time sweeps equal area farther away from the sun, planet moves slower closer to the sun, planet moves faster

36. Coordinates Systems in 3D Space Cartesian Coordinates:

37. Coordinates Systems in 3D Space Cylindrical Coordinates:

38. Coordinates Systems in 3D Space Spherical Coordinates: