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Unformatted text preview: 13.2. VECTORS Geometrical Approach to Vectors. The notion vector is used to indicate a quantity (like velocity, acceleration, force) that has a magnitude and a direction . It is represented by an arrow or a directed line segment. The length of the arrow is the magnitude and the arrow points the direction of a vector. We denote the vectors by using boldface letters u, v, a, b, etc. . The zero vector denoted by has length 0 and it the only vector with no specific direction. Each vector has many representatives and the most important from them is the position representative (the position vector) with initial point at the origin. Each representative of a vector that is an arrow has an initial point and a terminal point. Vector addition, sum of vectors. If u and v are vectors positioned as a chain such that the initial point of v is the terminal point of u , then the sum u + v is the vector from the initial point of u to the terminal point of v . Triangular rule, Parallelogram rule to sum vectors Commutative Law of the Summation of Vectors. Given vectors v 1 , v 2 ,..., v n . Then, the sum of these vectors does not depend on the order in which we perform the summation (on the order in which we construct the summation chain). Scalar Multiplication of Vectors. Given a scalar c and a vector v . The scalar multiple of v by c is a vector c v whose length is  c  times the length of v and a direction the same as v if c > 0 and opposite to v if c < 0. If c = 0 or v = then c v = . Difference of Vectors. The difference u v is defined by u v = u + ( v ) and this is trivially extended to more vectors. Algebraic Approach to Vectors. We start with a Cartesian planar (twodimensional) or threedimensional coordinate system. Given a vector a . Take its position representative (the position vector)→ OP with initial point at the origin. Then the coordinates of its terminal point P are ( a 1 ,a 2 ) or ( a 1 ,a 2 ,a 3 ), correspondingly. These coordinates are called the COMPONENTS of v and we write: a = h a 1 ,a 2 i ; a = h a 1 ,a 2 ,a 3 i . For vectors we use h a 1 ,a 2 ,a 3 i for vectors not to confuse with coordinate of a point ( a 1 ,a 2 ,a 3 ). Components of a Vector. Given the points A ( x 1 ,y 1 ,z 1 ) and B ( x 2 ,y 2 ,z 2 ). Then the vec tors v with a representative→ AB is v = h x 2 x 1 ,y 2 y 1 ,z 2 z 1 i . Note that a vector is completely determined by its components. Why? Magnitude (Length) of a Vector. Given a vector v . Its length is the length of any 1 of its representatives, including the position representative. The length of a planar, two dimensional vector a = h a 1 ,a 2 i is  a  = q a 2 1 + a 2 2 and the length of a threedimensional vector a = h a 1 ,a 2 ,a 3 i is  a  = q a 2 1 + a 2 2 + a 2 3 ....
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 Fall '09
 Dr.D.Dryanov
 Linear Algebra, Vectors, Dot Product

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