1.  References  Chapter 2.11 - 2.13, 9.2.1

2. DOT PRODUCT (2.11)

3. D.P. APPLICATION #1

4. PROBLEM* *: I’m intentionally not going to put problem solutions in these slides – take good notes!

5. DOT PRODUCT, CONT. α β γ A B C Law of Cosines is like the Pythagorean Theorem for any type of triangle (not just right triangles) Note: for right triangles, the last term is 0…

6. DERIVATION OF INTERP. #2 Law of Cosines θ “squaring” step1, using step3 D.P. follows distributive rule & step 5 (F.O.I.L) “Quod Erat Demonstrandum”, or “which had to be demonstrated”, or this to a mathematician…

7. NOT CONVINCED? θ θ is ~55 degrees

8. EXAMPLE, CONTINUED Theta is ~55 degrees. Interpretation#1: Interpretation#2: (we estimated the angle (it’s more like 55.8 degrees) and rounded off the lengths, otherwise they'd be identical)

9.  We can come up with an exact value for θ, given any two vectors using a little algebra and our two definitions of dot product. APPLICATION OF D.P #2 (CALCULATION OF Θ)

10. PROBLEM

11. PROBLEM (PICTURE) θ n

12. Acute θ is the angle between v and w. In each of these cases, think of what cos(θ) would be… APPLICATION OF DOT PRODUCT #3 v w θ v w θ v w θ v w θ v w θ θ≈45 θ≈12 0 θ≈18 0 θ≈90 NOTE: We never have to deal with the case of θ > 180. Why?? cos(45)=0.707 cos(120)=-0.5 cos(90)=0 cos(180)=-1 Obtuse Right.

13. D.P. APPLICATION #3

14. PROBLEM

15. APPLICATION #4 (PROJECTION)

16. θ

17. θ

18. PROBLEM

19.  It works even if they make an obtuse angle APPLICATION #4, CONT.

20. CROSS PRODUCT (2.12)

21. C.P. MNEMONIC #1 subtract these… add these…

22. C.P. MNEMONIC #2 Memorize Me Increase the subscripts by 1, “wrapping” around from z=>x

23. NOTE ABOUT PARALLEL VECTORS

24. CROSS PRODUCT, CONT. θ

25. DIRECTION from http://web.aeromech.usyd.edu.au/statics/doc/math3.htm

26. ADDITIONAL PROPERTIES θ θ the parallelogram viewed along the green arrow base height

27. PRACTICE PROBLEM Turned left Turned right x y

28. RAYS (9.2.1)