1. Comsats Institute of Information Technology (CIIT), Islamabad
Calculus-III
Lecture No 4 by
Dr. Umer Farooq

2. Outlines of Lecture 03 Dot Product
Direction Angles and Direction Cosines
Decomposition of a vector into Orthogonal Vectors

3. Dot Product and Projections
Example 4.1
Let and be the angles shown in the accompanying figure. Find the direction cosines of .
Solution:

4. Dot Product and Projections

5. Dot Product and Projections
Example 4.2
The accompanying figure shows a cube.
(a)Find the angle between the vectors and to the nearest degree.
(b)Make a conjecture about the angle between the vectors
and ,and confirm your conjecture by computing the angle.
Solution:

6. Dot Product and Projections

7. Dot Product and Projections

8. Dot Product and Projections
Work Done
The work W done on the object by a constant force of magnitude acting in the direction of motion over a distance d to be
force *distance
Furthermore, if we assume that the object moves along
A line from to , then , so that work can be expressed entirely in vector form as

9. Dot Product and Projections
Work Done
In the case where a constant force is not in the direction of motion, but rather makes an angle with the displacement vector, then we define the work done by to be

10. Dot Product and Projections
Example 4.3
A wagon is pulled horizontally by exciting a constant force of 10lb at an angle of with the horizontal. How much work is done in moving the wagon 50ft?
A force lb is applied to a point that moves on a line from to If distance is measured in feet, how much work is done.
Solution:

11. Dot Product and Projections

12. Dot Product and Projections

13. Cross Products
Determinants

14. Cross Products
Determinants

15. Cross Products Properties of Determinants:
If two rows in the array of a determinant are the same, then the value of the determinant is 0.
Interchanging two rows in the array of a determinant multiplies its value by -1.

16. Cross Products Definition:
If and are vectors in 3-space then the cross product is the vector defined by

17. Cross Products

18. Cross Products

19. Cross Products Algebraic Properties of Cross Product:
If and are vectors in 3-space then the following properties hold

20. Cross Products
Note:
(a)The cross product is defined only for vectors in 3-space, whereas the dot product is defined for vectors in 2-space and 3-space
(b) The cross product of two vectors is a vector, whereas the dot product of two vectors is a scalar

21. Cross Products Geometric Properties of the Cross Product:
If and are vectors in 3-space, then

22. Cross Products
Example 4.4
Find a vector that is orthogonal to both of the vectors and
Solution:

23. Cross Products Geometric Properties of the Cross Product:
Let and be non zero vectors in 3-space, and let be angle between them

24. Cross Products

25. Cross Products

26. Cross Products
Example 4.5
Find the area of the triangle that is determined by the points ,
and
Solution:

27. Cross Products

28. Cross Products Scalar Triple Product:
If , and are three vectors in 3-space then the scalar triple product

29. Cross Products

30. Cross Products Geometric Properties of Scalar Triple Product:
Let , and are non zero vectors in 3-space then

31. Cross Products
Example 4.5
Find two unit vectors that are normal to the plane determined by the points ,
and
Solution:

33. Cross Products
Example 4.5
Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector
Solution:

34. Cross Products

35. Vectors
Have a Nice Day
Thank You