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Unformatted text preview: Section 13.2 Vectors “Direction and Magnitude” 1. Basic Definitions Vectors are one of the main ideas used in multivariable calculus. A vector is a quantity which consists of a direction and a magnitude . The following are all examples: Velocity : direction is which way you are going and magnitude is speed. Force : direction is which was the force is being directed, magnitude is the amount of force Displacement : direction and distance of something moved from one point to another We shall initially deal with displacement vectors because they are easy to understand - however, all the results we shall consider are true of general vectors. There are a number of important definitions and some new terminology we need. ( i ) The displacement vector from point A to point B is an arrow with its tail at A and its tip at B . ( ii ) We call A the initial point and point B the terminal point and denote the vector by vector AB . ( iii ) The magnitude of the vector vector AB is defined to be the length and the direction is the direction in which it points. We denote its magnitude by || vector AB || . ( iv ) Two vectors are considered equivalent or equal if they have the same magnitude and direction (even if they have different initial and end positions). ( v ) The zero vector denotes vector 0 is defined to be the vector of mag- nitude 0. We define it to have no direction. For example, in the picture below, vector AB = vector EF but neither are equal to vector CD . A B C D E F We note that when representing a vector by a letter, we shall always include a bar or arrow over it to avoid confusion with scalars (numbers) 1 2 which are just magnitudes - the book does not do this!!! (So u is a number, but vectoru or ¯ u is a vector). 2. Operations on Vectors Just like with numbers, we can define basic arithmetic operations on vectors. There are three basic operations: ( i ) Addition If ¯ u and ¯ v are vectors so the initial point of ¯ v is the end point of ¯ u , we define the sum u + v to be the vector with the initial point the same as ¯ u and the end point the same as ¯...
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