1. Section 12.3 The Dot Product
MAT 1236 Calculus III
Section 12.3
The Dot Product

2. HW…
WebAssign 12.3
(19 problems, 108 min.)

3. Preview Define a new operation on vectors. Angle(s) between vectors
Projection of vectors
Formula
Why the formula makes sense? (More important in a long run)

4. We are Interested in … The angle between two forces
The component of one force along the direction of another force

5. Vectors
The angle between two forces

6. Dot Product
If and , the dot product of a and b is the number

7. Example 1
(a) (b)

8. Dot product and Length

9. Properties

10. Properties

11. Geometric Meaning

12. Formula

13. PPFNE
Derive the formula of the angle between two vectors

14. Example 2
Find the angle between the given vectors

15. Special Case: Orthogonal Vectors
Two vectors are orthogonal if the angle between them is a right angle.

16. Example 3
Find the value of x such that the given vectors are orthogonal.

17. Projection
Vector Projection Scalar Projection

18. Remark
Scalar Projection

19. Make Sense?
Vector Projection How do I know the vector is in the same direction of a?

20. Make Sense?
Vector Projection How do I know <2,2> is in the same direction of <1,1>?

21. Make Sense?
Vector Projection Does the “length” of the vector agree with what we know?

22. Proof: We need to recall…
Vector Projection

23. Unit Vectors A unit vector is a vector with length is 1
e.g. 1, 0 , 0, 1 , ,

24. Unit Vectors
The unit vector 𝑢 in the same direction as vector 𝑏 𝑢= 1 𝑏 𝑏

25. Unit Vectors
Unit vector 𝑢 in the same direction as vector 𝑏 𝑢= 1 𝑏 𝑏 So, 𝑏= 𝑏 𝑢 A vector equals to its length times the unit vector along the same direction

26. Proof:

27. Example 4
Find the vector projection of 𝑏 onto 𝑎.