1. Comsats Institute of Information Technology (CIIT), Islamabad
Calculus-III
Lecture No 3 by
Dr. Umer Farooq

2. Outlines of Lecture 02
Vector Addition, Subtraction and Scalar Multiplication
Unit Vector, Normalization of a Vector
Vectors Determined by Length and Angle
Resultant of Two and Three Forces

3. Dot Product and Projections
Definition:
If and are vectors in 2-space then the dot product between and is defined as
and is defined as
Similarly, if and are vectors in 3-Space then the dot product is defined as

4. Dot Product and Projections
Example 3.1
Find the dot product in the following cases.
Solution:

5. Dot Product and Projections
If , and are three vectors , then the following results hold

6. Dot Product and Projections
Theorem:
If and are vectors in 2-space or 3-space and if is the angle between them then
Proof:

7. Dot Product and Projections

8. Dot Product and Projections
Example 3.2
Find the angle between the vector and
(a) (b) (c)
Solution:

9. Dot Product and Projections

10. Dot Product and Projections

11. Dot Product and Projections
Perpendicular and Parallel Vectors:

12. Dot Product and Projections
Direction Angles In an xy-coordinate system, the direction of a non-zero vector is completely determined by the angles and between and the unit vectors and .
Fig.3.1

13. Dot Product and Projections
Direction Angles
Fig.3.2

14. Dot Product and Projections
Direction Cosines
In both 2-space and 3-space the angle between a non-zero vector and , and are called the direction angles of , and the cosines of those angles are called the direction cosines of If

15. Dot Product and Projections
Example 3.3
Find the direction cosines of the vector and approximate the direction angles to the nearest degree.
Solution:

16. Dot Product and Projections

17. Dot Product and Projections
Example 3.4
Find the angle between a diagonal of the cube and one of its edges.
Solution:

18. Dot Product and Projections
Decomposing Vectors into Orthogonal Vectors :
In many applications it is desirable to “decompose” a vector into a sum of two orthogonal vectors with convenient specified directions. For example fig. 3.3 shows an inclined plane. The downward force that gravity exerts on the block can be decomposed into the sum
Fig.3.3

19. Dot Product and Projections
Decomposing Vectors into Orthogonal Vectors :
where the force is parallel to the ramp and the force is perpendicular to the ramp. The forces
and are useful because is the force that pulls the block along the ramp and is the force that block exerts against the ramp.

20. Dot Product and Projections
Decomposing Vectors into Orthogonal Vectors :

21. Dot Product and Projections
Decomposing Vectors into Orthogonal Vectors :

22. Dot Product and Projections
Decomposing Vectors into Orthogonal Vectors :

23. Dot Product and Projections
Example 3.5
Let , and
Find the scalar and vector components of along and
Solution:

24. Dot Product and Projections

25. Dot Product and Projections
Example 3.6
A rope is attached to a 100 lb block on a ramp that is inclined at an angle of with the ground. How much force does the block exert against the ramp, and how much force must be applied to a rope in a direction parallel to the ramp to prevent the block from sliding down the ramp?
Solution:

26. Dot Product and Projections

27. Dot Product and Projections
Orthogonal Projections

28. Dot Product and Projections

29. Dot Product and Projections
Example 3.7
Find the orthogonal projection of on
and then find the vector component of orthogonal to .
Solution:

30. Dot Product and Projections

31. Dot Product and Projections
Example 3.8
Find so that the vector from the point to the point is orthogonal to the vector from to the point
Solution:

32. Dot Product and Projections

33. Dot Product and Projections
Example 3.9
Find two unit vectors in 2-space that make an angle of with
Solution:

34. Dot Product and Projections

35. Dot Product and Projections

36. Vectors
Have a Nice Day
Thank You