1.  References  Chapter 2.11, 2.13

2. DOT PRODUCT (2.11)

3. DOT-PRODUCT APPLICATION #1 (LENGTH-SQUARED)

4. IMPORTANT TIDBITS FROM CH. 2.13 distributive rule and vector-scalar multiplication distributive rule and vector-vector dot product

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

6. 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…

7. 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…

8. NOT CONVINCED? θ θ is ~55 degrees

9. 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)

10.  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)

11. PROBLEM

12. PROBLEM (PICTURE) θ n

13. 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.

14.  We can classify what type of angle is made by two vectors by looking at the sign of the dot product. Acute: Obtuse: Right: If v and w are both unit-length (normalized), we can make some more observations: APPLICATION OF DOT PRODUCT #3 (Θ “CATEGORIZATION”) if they are equal (the d.p is close to 1 if they’re in the same general direction). ALWAYS! if they are opposite (the d.p is close to -1 if they’re in generally opposite directions).

15. PROBLEM

16. APPLICATION #4 (PROJECTION)

17. θ

18. θ

19. PROBLEM

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