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Unformatted text preview: 1 Week 1 4.1 Vectors and Lines Many quantities that we measure can be completely defined by a real number called a scalar . ex. length, weight, volume, area Quantities that are defined by the two components: magnitude and direction are called vectors . ex. displacement, velocity, force A vector can be represented geometrically as a directed line segment. A is the initial point B is the terminal point and AB =- AB is the vector directed from A to B . The magnitude of- AB is the length of the arrow and we write k- AB k . We often use the lower case letters ~u,~v, ~w to refer to vectors and k , l and m to denote scalars. We can think of vectors as a new number system and we can do many familiar operations with vectors. Equality of Vectors Two vectors ~u and ~v are equal if they have the same magnitude and direction. Note: Position of a vector is not important. Sum To find the sum of two vectors ~u and ~v : Position ~v so its initial point lies on the terminal point of ~u . Then ~u + ~v is the vector from the initial point of ~u to the terminal point of ~v . 1 Note: ~u + ~v = ~v + ~u Zero Vector A vector of magnitude 0 is called the zero vector and is denoted by ~ 0. Note: For any vector ~v , we have ~v + ~ 0 = ~v . Negative Vector The vector- ~v has the same length as ~v but is oppositely directed. Note: ~v + (- ~v ) = ~ Scalar Multiplication If a 6 = 0 is a real number (scalar) and ~v 6 = ~ 0: 1. The magnitude of a~v is k a~v k = | a |k ~v k ....
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