Mars-Bound

by Evelyn Baldwin, Brigitte Wetz, and Liz Brown

Introduction Anchor Video Concept Map Project Calendar Lesson Plans Letter to Parents Assessments Resources Modifications Grant

Benchmark Lesson – Dot Product – Pre-Calculus LESSON PLAN – Dot Products and What Good Are Dot Products Good For? Name: Elizabeth Brown Title of Lesson: Dot Products Source of Lesson: From Drexel University by The Math Forum @ Drexel (http://mathforum.org/) Length of Lesson: 2 50 minutes lessons Description of the Class: Pre-calculus – Mars Rover Curriculum TEKS Address Pre-calculus- (6)  The student uses vectors to model physical situations. The student is expected to: (A)  use the concept of vectors to model situations defined by magnitude and direction; and (B)  analyze and solve vector problems generated by real-life situations. Overview: To explain the dot product and to show work (w) equals force (f) times distance (d) is a dot product. Also, to show why the dot product is useful. Performance Objectives The student will be able to: ·Perform the dot product calculation ·Determine the importance of dot product ·Relate the importance of the dot product to the idea of work Lecture: Definition of the Dot Product Given two vectors V and W, suppose they are represented by the coordinates a = (xa, ya) and b = (xb, yb). (Place tails of V and W at the origin. Then their heads are at a = (xa, ya) and b = (xb, yb).) Definition: The dot product of V and W is defined to be VW = xa*xb + ya*yb It's hard to see what the dot product is good for at this stage of the game. We'll work on it. One important thing ... What's VV? VV = xa^2 + ya^2, but that's just the length of V squared - VV = |V||V| Dot Product of Unit Vectors The dot product of unit vectors U and V Consider the triangle formed from U, V, and the vector W from the head of U to the head of V. We'll calculate the dot product by applying the Law of Cosines to the triangle formed from vectors U, V, and W. Law of Cosines: |W|^2 = |U|^2 + |V|^2 -2|U|*|V| cos t, where t is angle aOb. In our case, since U and V are unit vectors, this is simply |W|^2 = 1 + 1 - 2 cos t. What is this vector W? |W|^2 = |V-U|^2 = (V-U)(V-U) = VV - 2 UV + UU = 1 -2UV + 1. Since U + W = V, W = V - U. So, combining this expression for |W|2 with the one obtained from the Law of Cosines, we have 1 + 1 - 2 cos aOb = 1 -2 UV +1, hence, for unit vectors, UV = cos t, t the angle between them. Dot Products of Non-Unit Vectors Suppose V and W are not (necessarily) unit vectors. What is VW? Hint: What about V/|V| and W|W|? They're unit vectors. So what's V/|V|W|W|? Cos t. So what's VW? V/|V|W/|W| = Cos t, so VW = |V||W| Cos t. Big formula of the section: VW = |V||W| Cos t M which is true for all vectors, V and W, unit or not. What Good Are Dot Products Good For? The dot product formula is easy to compute and gives you information about the angle between vectors. Many times you don't need an exact angle - it's often enough to know whether or not two vectors are perpendicular . (How could you tell this from the dot product?) The dot product formula has to do with two vectors. In general two vectors determine a plane. The formula extends directly to vectors in three-dimensional space (and even higher!). Physicists think (happily) that the dot product tells you about work. If you have a constant force applied directly in the direction an object is moved, then the amount of work = (force)(distance). If there is a constant force applied at an angle to the direction an object is moved, then the amount of work = (force) (distance), since this gives the component of force in the direction the object is being moved times the distance moved.