September 23, 2018 Coming up: Mon 10/1: Exploration Topic Due! Wed 10/10: PSAT Fri 10/12: Vector Unit Exam (Ch 12 & 13) Fri 10/12: Begin Exploration writing Wed 10/31: Exploration Final Due! 1. Apply vector addition to real world problems 13A: #1 5 (Problem solving) Multiple forces or velocities influencing an object, add as vectors. As in flying objects... An airplane needs to travel due north at 300 mph to reach it's destination on time. There is a wind blowing from the SE at 50 mph. How fast and in what direction must the plane fly? 13A: #1 5 (Problem solving) 9/26 13A Problem solving 1
September 23, 2018 Coming up: Mon 10/1: Exploration Topic Due! Wed 10/10: PSAT Fri 10/12: Vector Unit Exam (Ch 12 & 13) Fri 10/12: Begin Exploration writing Wed 10/31: Exploration Final Due! UNM PNM 13A: #1 5 (Problem solving) Present #4 & 5 Parametric Equations 1. Understand and use parametric and vector equations 13B: #1 8 (2D & 3D Lines) QB: #1,10 (QB Practice) The two equations above describe a line. Notice that we can organize this in vector form: Vector Equation Let's look at this in more detail. Algebraically: Vector Equation of a Line r = p + td with t a real number r is a position vector to any point on the line p is the position vector of the reference point (t = 0) d is the direction vector which controls the slope t is the parameter (scalar) which controls distance from p in the direction of d. What is the slope of a line represented in vector form? Slope of a Line in Vector Form The slope of the line r = p + tv is controlled by the direction vector v Note the reversal! Always think meaning! Why is there more than one correct answer? How does the algebra work in 3D? 13B: #1 8 (2D & 3D Lines) QB: #1,10 (QB Practice) 9/28 13B 2D&3D Lines 2
13B: #1 8 (2D & 3D Lines) Discuss 7 & 8. Any others? QB: #1,10 (QB Practice) How is the standard form of a line related to the vector form of a line? 1. Find the angle between two lines. 13C: #1 5 (Angle between lines) 13D: #1 9 odd (Constant velocity problems) QB: #6,7,9 (QB Practice) Watch http://www.youtube.com/watch?v=0lg53 ogf2k (12:52) We know how to use dot products to find angles between vectors. How can we use this to find the angle between two lines? Recall: For lines extending indefinitely, we agree that: Thus we look at: The angle between two lines refers to the acute angle between them. Find the angle between the lines 3x 7y = 5 and y = 4x + 8 10/1 13C Angle bewteen lines 1
1. Use vector lines to solve problems involving constant velocity. The speed of a moving object represented by a vector equation is the magnitude (length!) of the direction vector. Note: Watching the YouTube video will prepare you well for Monday's lesson. 13C: #1 5 (Angle between lines) 13D: #1 9 odd (Constant velocity problems) QB: #6,7,9 (QB Practice) Watch http://www.youtube.com/watch?v=0lg53 ogf2k (12:52) 10/1 13D Constant Velocity 2
13C: #1 5 (Angle between lines) Present 3,5 13D: #1 9 odd (Constant velocity problems) Present 7,9 QB: #6,7,9 (QB Practice) Present Head's Up: Today and Friday are the last two lessons in this unit. Next week we will review and spend some time on PSAT (no class Wed) The unit exam on vectors will be on Friday, 10/12 Monday 10/15 we will continue work on your Exploration NOW (as in this week!) is the time to research A video of topics in this lesson is available at: http://www.youtube.com/watch?v=0lg53 ogf2k 12:52 1. Use vectors to find the distance from a point to a line. 13E: #1 7 (Distance from line to point) QB: #3,4 (QB Practice) The "distance" from a point to a line refers to the shortest distance. Thus, we are interested in the perpendicular path from the point to the line. The key is to recall that perpendicular directions create a zero dot product. So, in short we need to find the length of the perpendicular vector from the point to the line. L N A(5, 3) P (6, 5) Don't try to remember how to do this. Understand your tools. Some examples 13E: #1 7 (Distance from line to point) QB: #3,4 (QB Practice) 10/3 13E Distance from a Point to a Line 3
Alternate Method for the Distance from a Point to a Line Based on the dot product as a projection of one vector onto another The key geometric idea: L A dot product is the projection of one vector onto the other. N A(5, 3) P (6, 5) The negative sign just means that we drew it wrong - N is to the left of A. Don't try to remember how to do this. Understand your tools. Alternate Distance from Point to a Line 4
13E: #1 7 (Distance from line to point) Present #1,3,5,7 Ex 2: #17.12,19.6,20.1,46.1 Quick Quiz M1A1 A1 M1 set a dot product to 0 A1 5. One point is (0, 7) A1A1 6. The set of points is the line 5x + 2y = 14 A1 Or any point on [4, 3] + t[2, 5] or equivalent 7. [3, 4] is perpendicular to the line A2 1. Use vectors to find whether and when two lines intersect. 13F: #1 5 (Intersecting lines) 13G: #1 all (Multiple lines 2D & 3D) QB: #2,5,8 (QB Vectors) What does it mean for two lines to intersect? Intersecting lines Two lines intersect at a point where all their coordinates are equal. The general coordinates of a line are the parametric equations of the line. In general you need to use different parameters for the different lines In 2 dimensions, it's straightforward. What about in 3 dimensions? Two lines can intersect, but it doesn't mean that two objects travelling along the lines will collide. Why not? What is required for a collision? 10/513F Intersecting Lines 5
1. Use vectors to find whether and when two lines intersect. Let's summarize the possible relationships between two lines: Lines in 2 Dimensions Up to now, we have solved using techniques for a "linear system": Substitution, Elimination, Matrix Inversion (not an SL topic but allowable and helpful if you have a calculator) Lines in 3 Dimensions Coplanar Lines Lines in the same plane are either: Intersecting Parallel Coincident Skew Lines Lines in different planes cannot: Intersect Be parallel Be coincident The angle between them is as if one were translated to be in the same plane as the other. Suggestion: Do this by Monday 10/8 Do this by Wed 10/10 Review next time, Unit test Friday 10/12 13F: #1 5 (Intersecting lines) 13G: #1 all (Multiple lines 2D & 3D) QB: #2,5,8 (QB Vectors) 10/5 13G Relationships between lines 6
13F: #1 5 (Intersecting lines) 13G: #1 all (Multiple lines 2D & 3D) QB: #2,5,8 (QB Vectors) Some key points: Pay attention to direction, particularly with sums & differences To find angle between vectors they must be tail to tail Have your formula sheet there may be connections to other ideas Your unit exam will be on Friday 10/12 Review 12C: #1 15 (Review) Review 13C: #1 7 all (Review) QB not done 10/8 Review work in class 7
2:12.7 Intro to parametric equations 2:11.9 2:20.8 20.10 Ex: 3.15.3 Dot products revealed Ex: 3.9.7 Dot product distributivity The dot product of u = [a, b] and v = [c, d] is defined as ac + bd. In three dimensions: The dot product of u = [a, b, c] and v = [d, e, f] is defined as ad + be + cf. Properties: When u v = 0, u and v are perpendicular. Try some practice: u = [ 2, 3, 1] v = [0, 1, 2] w = [1, 2, 1] Find: 4u Answer: [ 8, 12, 4] u + v Answer: [ 2, 4, 3] 4u v Answer: [ 8, 11, 2] u (v + w) Answer: 8 u v + u w Answer: 8 What do these two results suggest? Can you prove it? Ex: 3.10.11 Commutivity, other properties Let u = [a, b, c] v = [p, q, r] and w = [k, m, n] (a) Verify that u v = v u (b) What is the significance of u u? (c) What does u v = 0 tell us about u and v? (d) Is it true that u (v + w) = u v + u w for all vectors u, v, and w? (e) If u and v represent sides of a ogram, then u + v and u v represent the diagonals. What does (u + v) (u v) = 0 tell us about the ogram? Ex: 3.11.10 3D magnitude Let u = [a, b, c]. The notation u represents the magnitude or length of u. We calculate u from Notice that u u = a 2 + b 2 + c 2. This is the same as u 2. A very important relationship! Ex: 3.11.11 Is it true that (u v) (u v) = u u 2u v + v v? Note the similarity with (a b) 2. Use dot products to find the angle between vectors: Let u = [3, 4] and v = [ 5, 12]. Another form of the vector law of cosines is So... the angle between u and v is given by (often you'll need a calculator to get a value. When do you not need one?) Exeter problems 8