1. © 2013 Pearson Education, Inc.

2. Let’s just start with the Dot Product formula 12J The scalar product is a manner of multiplying vectors. The scalar product of two vectors can be constructed by: taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector.

3. © 2013 Pearson Education, Inc. The Definition of the Dot Product of Two Vectors The dot product of u = and v = is Ex.’s Find each dot product. Let’s just start with the Dot Product formula 12J

4. © 2013 Pearson Education, Inc. Basically, the scalar product (A.K.A. the Inner Product) 12J Read as, “V dot W”

5. © 2013 Pearson Education, Inc. If you want to know more about the dot product check out the videos on the IB class website 12J

6. © 2013 Pearson Education, Inc. What does this mean? 12J Normal multiplication combines growth rates “3 x 4″ can mean “Take your 3x growth and make it 4x larger (i.e., 12x)”. A vector is “growth in a direction”. The dot product lets us apply the directional growth of one vector to another The result is how much we went along the original path (positive progress, negative, or zero).

7. © 2013 Pearson Education, Inc. What does this mean? 12J Seeing Numbers as vectors Let’s start simple, and see 3 x 4 as a dot product : 3 is “directional growth” in a single dimension (x-axis, let’s say) 4 is “directional growth” in that same direction. 3 x 4 = 12 means 12x growth in that single dimension. Ok?

8. © 2013 Pearson Education, Inc. What does this mean? 12J Seeing Numbers as vectors Let’s start simple, and see 3 x 4 as a dot product : Suppose each number refers to a different dimension. 3 means “triple your bananas” (sigh… or “x-axis”) 4 means “quadruple your oranges” (y-axis). They’re not the same type of number: what happens when we apply growth, aka use the dot product, in our “bananas, oranges” universe? ”

9. © 2013 Pearson Education, Inc. What does this mean? 12J Seeing Numbers as vectors Let’s start simple, and see 3 x 4 as a dot product : (3,0) is “Triple your bananas, destroy oranges” (0,4) is “Destroy your bananas, quadruple oranges

10. © 2013 Pearson Education, Inc. What does this mean? 12J Seeing Numbers as vectors Let’s start simple, and see 3 x 4 as a dot product : Applying (0,4) to (3,0) means “Destroy banana growth, quadruple orange growth”. But (3, 0) had no orange growth to begin with The net result is 0 (“Destroy all your fruit, buddy”).

11. © 2013 Pearson Education, Inc. What does this mean? 12J See how we’re “applying” and not adding? With addition, we sort of smush the items together: (3,0) + (0, 4) = (3, 4) [a vector which triples your oranges and quadruples your bananas]. “Application” is different. We’re mutating the original vector according to the rules in the second. And the rules are “Destroy your banana growth rate, and triple your orange growth rate“. And, sadly, this leaves us with nothing.

12. © 2013 Pearson Education, Inc. 12J Mario-Kart Speed Boost In Mario Kart, there are “boost pads” on the ground that increase your speed (Never played? I’m sorry.)

13. © 2013 Pearson Education, Inc. 12J Imagine the red vector is your speed (x & y direction), and the blue vector is the orientation of the boost pad (x & y direction). Larger numbers are more power. How much boost will you get? For the analogy, imagine the pad multiplies your speed:

14. © 2013 Pearson Education, Inc. 12J If you come in going 0, you’ll get nothing [if you are just dropped onto the pad, there’s no boost] If you cross the pad perpendicularly, you’ll get 0 [just like the banana obliteration, it will give you 0x boost in the perpendicular direction]

15. © 2013 Pearson Education, Inc. 12J But, if we have some overlap, our x-speed will get an x-boost, and our y-speed gets a y-boost:

16. © 2013 Pearson Education, Inc. What does this mean? 12J The final result of this process can be: Zero: we don’t have any growth in the original direction Positive Number: we have some growth in the original direction Negative Number: we have negative (reverse) growth in the original direction

17. © 2013 Pearson Education, Inc. Better? 12J

18. © 2013 Pearson Education, Inc. So, you’re sitting there and you ask yourself, “Self, how do I find the angle between vectors?” 12J Great Question!

19. © 2013 Pearson Education, Inc. Consider the following: 12J v w Translate one of the vectors so that they both start at the same point

20. © 2013 Pearson Education, Inc. Consider the following: 12J This is the vector: -v + w = w - v v w θ Has length: |w – v|

21. © 2013 Pearson Education, Inc. Consider the following: 12J v w θ |w – v| From the Law of Cosines where c is the side opposite the angle theta:

22. © 2013 Pearson Education, Inc. However, 12J v w θ |w – v|

23. © 2013 Pearson Education, Inc. 12J

24. © 2013 Pearson Education, Inc. 12J

25. © 2013 Pearson Education, Inc. 12J So, to find the angle between vectors can be found using:

26. © 2013 Pearson Education, Inc. 12J Algebraic Properties of the Scalar Product

27. © 2013 Pearson Education, Inc. 12J Other Properties of the Scalar Product If v and w are perpendicular or “orthogonal” If v and w are non-zero parallel vectors

28. © 2013 Pearson Education, Inc. HOMEWORK Test Next Friday Over chapter 12 and Two-ish problems from chapter 6

29. © 2013 Pearson Education, Inc. HOMEWORK page 310(12j) Numbers 1 – 23 Skip #9 Review 12A and 12B On 12B (Skip #12)