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# Week_1 - 1 Week 1 4.1 Vectors and Lines Many quantities...

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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 -→ AB . 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

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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 ) = 0 Scalar Multiplication If a = 0 is a real number (scalar) and v = 0: 1. The magnitude of av is av = | a | v .
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