### Course 4 Unit 2 - Vectors and Motion ©2010

In Course 4: Preparation for Calculus, geometry and algebra become increasingly intertwined. Students develop understanding of two-dimensional vectors and their application and the use of parametric equations in modeling linear, circular, and other nonlinear motion. In addition, students intending to pursue programs in the mathematical, physical, and biological sciences, or engineering extend their ability to visualize and represent three-dimensional surfaces using contours, cross sections, and reliefs; and to visualize and sketch surfaces and conic sections defined by algebraic equations. They also extend their understanding of, and ability to reason with, trigonometric functions to prove or disprove trigonometric identities and to solve trigonometric equations. They also geometrically represent complex numbers and apply complex number operations to find powers and roots of complex numbers expressed in trigonometric form. (See the CPMP Courses 1-4 descriptions.)

Unit Overview
Motion is a frequently occurring aspect of our lives. It is natural to ask how motion can be modeled mathematically. Two mathematical tools are introduced in this unit that help students model motion. The first is the vector, which is introduced initially as a free vector (directed line segment) that is drawn wherever it may be needed and later as a position vector attached to a specific point. It is the position vector representation that leads to the introduction of the second tool, which is parametric representation of locations specified by vectors. The set of these locations forms the graph of the motion as a function of a third variable, usually time. The parametric representations and a graphing calculator or computer graphing software allow students to see linear, projectile, circular, and elliptical motions of objects and to analyze the resulting paths.
Lesson 1 develops vectors and their geometric representation, scaling of vectors (including the opposite of a vector), and combining vectors by addition. These concepts are developed in the context of navigation and applied to other situations. In Lesson 2, vectors are analyzed in a coordinate system. The coordinate representation of vectors is then used to prove geometric linear motion algebraically using parametric equations. Lesson 3 extends simulation of motion using parametric equations to model projectile motion and circular and elliptical motion.

 Objectives of the Unit Describe and use the concept of vector in mathematical, scientific, and everyday situations Represent vectors geometrically and operate on geometric vectors Describe, represent, and use vector components and operations synthetically and analytically Investigate and justify general properties of vectors and vector operations Provide vector proofs of properties of triangles and parallelograms Use vector concepts to parametrically represent linear, projectile, circular, and elliptical motions in a plane Analyze motions using parametric models

Sample Overview
The sample investigation below is Investigation 2 from Lesson 2. In the first investigation of this lesson, students explored coordinate representations of vectors. In this investigation, after exploring operations on vectors, students examine how vectors can be used to establish geometric relationships.

Instructional Design
Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.