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Introduction to Vector Addition and Scalar Multiplication

Introduction to Vector Addition and Scalar Multiplication -...

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Introduction to Vector Addition and Scalar Multiplication In the strictly mathematical definition of a vector, the only operations that vectors are required to possess are those of addition and scalar multiplication. (Compare this with the operations allowed on ordinary real numbers, or scalars, in which we are given addition, subtraction, multiplication, and division). For instance, in a raw vector space there is no obvious way to multiply two vectors together to get a third vector--even though we will define a couple of ways of performing vector multiplication in Vector Multiplication . It makes sense, then, to begin studying vectors with an investigation of the operations of vector addition and scalar multiplication. This section will be entirely devoted to e xplaining addition and scalar multiplication of two- and three-dimensional vectors. This explanation will involve two different, yet equivalent, methods: the
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Unformatted text preview: component method and the graphical method . The Component Method for Vector Addition and Scalar Multiplication When we mentioned in the introduction that a vector is either an ordered pair or a triplet of numbers we implicitly defined vectors in terms of components. Each entry in the 2-dimensional ordered pair ( a , b ) or 3-dimensional triplet ( a , b , c ) is called a component of the vector. Unless otherwise specified, it is normally understood that the entries correspond to the number of units the vector has in the x , y , and (for the 3D case) z directions of a plane or space. In other words, you can think of the components as simply the coordinates of the point associated with the vector. (In some sense, the vector is the point, although when we draw vectors we normally draw an arrow from the origin to the point.) Figure %: The vector (a, b) in the Euclidean plane....
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