1. Chapter 12 – Vectors and the Geometry of Space
12.3 – The Dot Product
12.3 – The Dot Product

2. Definition – Dot Product
Note: The result is not a vector. It is a real number, a scalar. Sometimes the dot product is called the scalar product or inner product.
12.3 – The Dot Product

3. Example 1 – pg.806 # 8 Find a  b a = 3i + 2j - k b = 4i + 5k
12.3 – The Dot Product

4. Properties of the Dot Product

5. Theorem – Dot Product
The dot product can be given a geometric interpretation in terms of the angle  between a and b.
12.3 – The Dot Product

6. Applying Law of Cosines
We can apply the Law of Cosines to the triangle OAB and get the following formulas:
12.3 – The Dot Product

7. Corollary – Dot Product
12.3 – The Dot Product

8. Example 2 – pg. 806 # 18
Find the angle between the vectors. (First find an exact expression then approximate to the nearest degree.) a = <4, 0, 2> b = <2, -1, 0>
12.3 – The Dot Product

9. Orthogonal Vectors
Two nonzero a and b are called perpendicular or orthogonal if the angles between them is  = /2.
12.3 – The Dot Product

10. Hints
The dot product is a way of measuring the extent to which the vectors point in the same direction.
If the dot product is positive, then the vectors point in the same direction.
If the dot product is 0, the vectors are perpendicular.
If the dot product is negative, the vectors point in opposite directions.
12.3 – The Dot Product

11. Visualization
The Dot Product of Two Vectors
12.3 – The Dot Product

12. Example 3
For what values of b are the given vectors orthogonal? <-6, b, 2> <b, b2, b>
12.3 – The Dot Product

13. Definition – Directional Angles
The directional angles of a nonzero vector a are the angles , , and  in the interval from 0 to pi that a makes with the positive axes.
12.3 – The Dot Product

14. Definition – Direction Cosines
We get the direction cosines of a vector a by taking the cosines of the direction angles. We get the following formulas
12.3 – The Dot Product

15. Continued
12.3 – The Dot Product

16. Example 4 pg. 806 #35
Find the direction cosines and direction angles of the vector. Give the direction angles correct to the nearest degree.
i – 2j – 3k
12.3 – The Dot Product

17. Definition - Vector Projection
If S is the foot of the perpendicular from R to the line containing , then the vector with representation is called the vector projection of b onto a and is denoted by projab. (think of it as a shadow of b.)
12.3 – The Dot Product

18. Definition continued
12.3 – The Dot Product

19. Visualization
Vector Projections
12.3 – The Dot Product

20. Definition – Scalar Projection
The scalar projection or component of b onto a is defined to be the signed magnitude of the vector projection, which is the number |b|cos, where  is the angle between a and b. This is denoted by compab.
12.3 – The Dot Product

21. Definition continued
12.3 – The Dot Product

22. Example 5 – pg807 #42
Find the scalar and vector projections of b onto a.
a = <-2, 3, -6>
b = <5, -1, 4>
12.3 – The Dot Product

23. More Examples
The video examples below are from section 12.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 1
Example 3
Example 6
12.3 – The Dot Product

24. Demonstrations Feel free to explore these demonstrations below.
The Dot Product
Vectors in 3D
Vector Projections
12.3 – The Dot Product