1. Vectors and the Geometry of Space
The Dot Product of Two Vectors
Written by Richard Gill
Associate Professor of Mathematics
Tidewater Community College, Norfolk Campus, Norfolk, VA
With Assistance from a VCCS LearningWare Grant

2. The Definition of Dot Product
In our first two lessons on vectors, you have studied:
Properties of vectors;
Notation associated with vectors;
Vector Addition;
Multiplication by a Scalar.
In this lesson you will study the dot product of two vectors. The dot product of two vectors generates a scalar as described below.
The Definition of Dot Product
The dot product of two vectors
The dot product of two vectors

3. Properties of the Dot Product
The proof of property 5.
Proofs of other properties are similar.

4. Consider two non-zero vectors
Consider two non-zero vectors. We can use their dot product and their magnitudes to calculate the angle between the two vectors. We begin with the sketch.
From the Law of Cosines where c is the side opposite the angle theta:

5. From the previous slide
Substituting from above
Simplifying

6. You have just witnessed the proof of the following theorem:
Example 1
Find the angle between the vectors:
Solution

7. Example 2
Solution
True or False? Whenever two non-zero vectors are perpendicular, their dot product is 0.
Congratulate yourself if you chose True!
Think before you click.

8. True or False? Whenever two non-zero vectors are perpendicular, their dot product is 0.
This is true. Since the two vectors are perpendicular,
the angle between them will be
True or False? Whenever you find the angle between
two non-zero vectors the formula
will generate angles in the interval

9. True or False? Whenever you find the angle between
two non-zero vectors the formula
will generate angles in the interval
This is False. For example, consider the vectors:
Finish this on your own then click for the answer.

10. True or False? Whenever you find the angle between
two non-zero vectors the formula
will generate angles in the interval
This is False. For example, consider the vectors:
When the cosine is negative the angle between the two vectors is obtuse.

11. An Application of the Dot Product: Projection
The Tractor Problem: Consider the familiar example of a heavy box being dragged across the floor by a rope. If the box weighs 250 pounds and the angle between the rope and the horizontal is 25 degrees, how much force does the tractor have to exert to move the box?
Discussion: the force being exerted by the tractor can be interpreted as a vector with direction of 25 degrees. Our job is to find the magnitude.

12. The Tractor Problem, Slide 2: we are going to look at the force vector as the sum of its vertical component vector and its horizontal component. The work of moving the box across the floor is done by the horizontal component. x y
Conclusion: the tractor has to exert a force of lbs before the box will move.
Hint: for maximum accuracy, don’t round off until the end of the problem. In this case, we left the value of the cosine in the calculator and did not round off until the end.

13. The problem gets more complicated when the direction in which the object moves is not horizontal or vertical. x y

14. A Vocabulary Tip:
When two vectors are perpendicular we say that they are orthogonal.
When a vector is perpendicular to a line or a plane we say that the vector is normal to the line or plane. x y
The following theorem will prove very useful in the remainder of this course.

15. (5,9)
(6,3)

16. Work is traditionally defined as follows: W = FD where F is the constant force acting along the line of motion and D is the distance traveled along the line of motion.
Example: an object is pulled 12 feet across the floor using a force of 100 pounds. Find the work done if the force is applied at an angle of 50 degrees above the horizontal.
100 lb
12 ft
Solution A: using W = FD we use the projection of F in the x direction.

17. Two Ways to Calculate Work
Example: Find the work done by a force of 20 lb acting in the direction N50W while moving an object 4 ft due west.

18. The last topic of this lesson concerns Direction Cosines.
We can calculate the direction cosines by using the unit vectors along each positive axis and the dot product.

19. For comments on this presentation you may email the author Professor Richard Gill
or the publisher of the VML Dr. Julia Arnold at