Welcome to Math 275 Topic 1-1: Introduction to Vectors

Transcription

1 Welcome to Math 275 Topic 1-1: Introduction to Vectors Textbook: Sections 12.1, 12.2 Warm-Up: Pythagorean Theorem The Pythagorean Theorem says what about a right triangle with legs of length a and b, and hypotenuse of length c? Warm-Up: Pythagorean Identity cos 2 θ + sin 2 θ =?

2 What Is This Class About? Single variable calculus (Calc I and II) deals with one-dimensional problems. In multivariable and vector calculus (Calc III), we will learn how to work on problems in higher-dimensional space primarily, the plane (R 2 ) and 3-space (R 3 ). In these problems, it is often the case that we need to take into account not only a number, but also a direction. For example, think of a force, which has both strength and direction; or velocity, which accounts for both speed and direction. Vectors are the mathematical tools that we use in these situations. Big Ideas Vectors have direction and magnitude (length). Vectors can be represented by: Arrows directed line segments from one point to another point. (geometric representation) Components ordered sets of numbers. (algebraic representation) Vectors scalars (numbers) points. Introduction to Vectors For this class, a vector (denoted v or v) is a mathematical object possessing both magnitude (also called length) denoted v and direction. Vectors are often represented as directed line segments (arrows). For this class, a scalar is a real number: c is a scalar if c R.

3 Points In Cartesian (Rectangular) Coordinates Fix the origin: In R 2 (the plane), points can be represented as ordered pairs of numbers: P = (x 0, y 0 ). In R 3 (3-space), points can be represented as ordered triples of numbers: P = (x 0, y 0, z 0 ). Each coordinate represents the distance along a coordinate axis. The sign of the coordinate represents the direction along the axis relative to the origin O = (0, 0, 0). The textbook uses the notation P(x 0, y 0 ) and P(x 0, y 0, z 0 ) for points. Vectors in Component Form Coordinates allow us to write vectors in component form. The position vector r that begins at the origin O = (0, 0, 0) and ends at the point P = (x, y, z) has component form: r = x, y, z The displacement vector PQ that begins at the point P = (x 1, y 1, z 1 ) and ends at the point Q = (x 2, y 2, z 2 ) has component form: PQ = x 2 x 1, y 2 y 1, z 2 z 1. The point where a vector starts is called the initial point of the vector. The point where a vector ends is called the terminal point of the vector. Notation: Vectors vs. Points Observe: Vectors are denoted using arrows or boldface, and angle brackets: v = x, y, z or v = x, y, z. Points are denoted without arrows or boldface, and using curved brackets: P = (x, y, z) or P(x, y, z).

4 Equivalent Vectors Two vectors v and w are equivalent, that is: if they have: v = w The same component form (algebraic representaion), or: The same magnitude and direction (geometric description. Course Website Bookmark the url: Contact information for Dr. Ultman and Josie Derrick (LA). Office hour and help session times (TBA). Links to log-in for WebAssign. Projected schedule, with links to slides, notes, and exam reviews. Syllabus; information on the textbook, WebAssign, and clickers and ResponseWare; links to resources; etc. Blackboard Blackboard will be used only for the following: Registering clickers (needs to be done by beginning of class on Friday 26 August). Posting exam and clicker scores, and (before the final exam) grade estimates. Sending s to members of the class. Discussion board (?). Blackboard is not set up to compute your final course grade.

5 WebAssign WebAssign will be used for homework. Current scores for problems sets and homework will be posted on WebAssign. To enroll for WebAssign, use the class key: boisestate There is a two week free trial period beginning Sunday 21 August. After this, you will need to purchase access. WebAssign, Continued Set notifications to alert you when assignments are due (link at top right of page). Check the instructions and directions at the top of each assignment. If you have a problem with WebAssign (for example, a problem is being marked incorrect and you can t figure out why), or a question about an assignment, use the ask your teacher link to send Dr. Ultman an . Contact Dr. Ultman before you use your last attempt! Extensions can be requested automatically after the due date. You cannot request an extension after you look at the answer key. Homework Settings Extensions are granted automatically, and available for 1 week after the assignment due date. Problems completed before due date get full credit. Problems completed after due date get 75% credit. Number of attempts may vary. Be sure to check each problem for number of attempts allowed. Practice anther version will be available for most daily problem sets. For other types of homework, it may or may not be available. For more details on homework, see the course syllabus: syllabus/