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

2. Orthogonal Vectors Angular Dependence

3. Scalar Product Scalar Product of a Vector with itself ? A. A = | A || A | cos 0 º = A 2

4. Scalar Product Scalar Product of a Vector and Unit vector ? x. A = | x || A | cosα = A x 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

17. 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/mu lticalc/coords/index.htmlhttp://www.flashandmath.com/mathlets/mu lticalc/coords/index.html

22. Line Integrals

26. Tutorial Evaluate: Where C is right half of the circle : x 2 +y 2 =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: …. Q.NO 2: Evaluate were C is the line segment from to