1. Scalar Product Scalar / Dot Product of Two Vectors Product of their magnitudes multiplied by the cosine of the angle between the Vectors

2. Orthogonal Vectors Angular Dependence

3. Scalar Product Scalar Product of a Vector with itself ? A . A = |A||A| cos 0º = A2

4. Scalar Product Scalar Product of a Vector and Unit vector ? x . A =|x||A|cosα = Ax Yields the component of a vector in a direction of the unit vector Where alpha is an angle between A and unit vector x ^ ^

5. Scalar Product Scalar Product of Rectangular Coordinate Unit vectors? x.y = y.z = z.x = ? = 0 x.x = y.y = z.z = ? = 1

6. Scalar Product Problem 3: A . B = ? ( hint: both vectors have components in three directions of unit vectors)

7. Scalar Product Problem 4: A = y3 + z2; B= x5 + y8 A . B = ?

8. Scalar Product Problem 5: A = -x7 + y12 +z3; B = x4 + y2 + z16 A.B = ?

9. Line Integrals

10. Line Integrals

11. Line Integrals

12. Line Integrals

13. Line Integrals

14. Line Integrals

15. Line Integrals

16. Line Integrals

17. Spherical coordinates

18. Spherical coordinates

19. Spherical Coordinates For many mathematical problems, it is far easier to use spherical coordinates instead of Cartesian ones. In essence, a vector r (we drop the underlining here) with the Cartesian coordinates (x,y,z) is expressed in spherical coordinates by giving its distance from the origin (assumed to be identical for both systems) |r|, and the two angles   and   between the direction of r and the x- and z-axis of the Cartesian system. This sounds more complicated than it actually is:   and    are nothing but the geographic longitude    and latitude. The picture below illustrates this

20. Spherical coordinate system

21. Simulation of SCS • http://www.flashandmath.com/mathlets/multicalc/coords/index.html

22. Line Integrals

23. Line Integrals

24. Line Integrals

25. Line Integrals

26. Tutorial • Evaluate: Where C is right half of the circle : x2+y2=16 Solution We first need a parameterization of the circle.  This is given by, We now need a range of t’s that will give the right half of the circle.  The following range of t’s will do this: Now, we need the derivatives of the parametric equations and let’s compute ds:

27. Tutorial ……… • The line integral is then :

28. Assignment No 3 • Q. No. 1:  Evaluate  where C is the curve shown below.

29. Assignment No 3: …. were C is the line segment from   to • Q.NO 2: Evaluate