Chapter9 - : Pre-Calculus - Chapter 9A Chapter 9A - A Brief...

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© : Pre - Calculus - Chapter 9A Chapter 9A - A Brief Description As life becomes more complicated, entities arise that not only have a magnitude , but also a direction associated with them. Such objects are called vectors . A common example of this occurs when you give someone directions to get to your home: go 5 blocks west, then 16 blocks north, and my house is on the corner. There are two vectors here. The first has a magnitude of 5 units and a direction, west; the second has a magnitude of 16 and a direction, north. These verbal descriptions of a vector are too cumbersome, and a better notation has been invented. It is the same notation which is used to locate points in the plane, R 2 or in three-space, R 3 . Representing Vectors . Example 1 : A vector which is one unit long and points due north can be represented by the ordered pair 1,0 , and a vector which has magnitude 5 and points northwest is 5 2 , 5 2 The individual values are called components . Since these vectors have only two components, we think of them as belonging to or lying in a plane. We often picture vectors as arrows. The tail of the arrow is at the origin and the arrow tip is located at the point whose Cartesian coordinates are the same as the ordered pair (triple) which represents the vectors. The vectors 1,1 , 4,3 ,and 3, 5 are plotted below. -5 -4 -3 -2 -1 0 4 5 - 5- 4 - 3 - 2 - 1 12345 (1,1) (-4,3) (3,-5) x y Two dimensions In R 2 (the plane) there are two special vectors. The two that are parallel to the coordinate axes. They are commonly denoted by the symbols i and j or by the pair of symbols e 1 and e 2 .Thatis , e 1 i e 2 j 0,1 -1 0 1 -1 1 x y 1 e 2 e © : Pre - Calculus
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© : Pre - Calculus - Chapter 9A Three dimensions In R 3 we have three special vectors which point in directions parallel to the coordinate axes. e 1 i 1,0,0 e 2 j 0,1,0 e 3 k 0,0,1 1 1 x y z 1 e 2 e 3 e There is an ambiguity in this notation. What does e 2 mean? Is it the ordered pair of numbers 0,1 or the ordered triple of numbers . The answer is determined by the context of the particular discussion in which this symbol appears. Coordinate vectors These vectors { e 1 , e 2 } are sometimes referred to as the coordinate vectors. That is, in R 2 the standard coordinate vectors are e 1 and e 2 . Question : What are the coordinate vectors of R 3 ? Answer: The coordinate vectors in R 3 are: e 1 , e 2 ,and e 3 Example 2 : Suppose you tell someone the way to get to your house is to go 7 blocks east ,then1 block south, and finally 4 blocks east. One way to indicate this in vector notation is 7,0 0, 1 4,0 Or using the coordinate vectors 7 e 1 e 2 4 e 1 11 e 1 e 2 Example 3 : Suppose you were given the following directions: 1,1 0, 4 3,0 0, 2 .
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This note was uploaded on 02/23/2010 for the course CHEM 107 taught by Professor Generalchemforeng during the Fall '07 term at Texas A&M.

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Chapter9 - : Pre-Calculus - Chapter 9A Chapter 9A - A Brief...

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