Vector_Tensors

Vector_Tensors - A Primer on Vectors, Basis Sets and...

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A Primer on Vectors, Basis Sets and Tensors ü Copyright Brian G. Higgins (2004) Vectors in ! 2 An ordered pair is a sequence of real numbers or scalars: (1) x = 8 x 1 , x 2 < We say the scalar x i is the i-th component of the ordered pair. The set of all ordered pairs is denoted by ! 2 . The ordered pair x is also referred to as a plane vector. Some examples of plane vectors include velocity, friction, force (when these concepts are restricted to the 2-D plane). As we shall see shortly vectors have both direction and magnitude , whereas scalars have just magnitude. A vector is usually represented geometrically as a directed line segment- an arrow- pointing in the direction of the vector, with a length that corresponds to its magnitude. The arrow head defines the terminal point of the vector. The base of the arrow defines the initial point of the vector. When one considers vectors in the plane, they are usually defined with respect to a specified coordinate system. For example, if we take the rectangular Cartesian coordinate system, then the vectors (2) a = 8 2, 3 < , b = 8 - 2, 3 < are shown schematically as Figure 1 -3 1 2 3 x -4 -3 -2 1 2 3 4 y a b When the base of each vector is located at the origin of the coordinate system, 8 0, 0 < , then the first component of the ordered pair (e.g., in the case of a , this is the scalar 2 ) is called the x-component of the vector, and the second component is called the y-component of the vector. Note that there is a one-to-one correspondence between all vectors in the plane and ordered pairs.
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Addition of vectors, and multiplication of a vector by a scalar are defined in terms of their components by the following rules: If a = 8 a 1 , a 2 < , b = 8 b 1 , b 2 < are any two plane vectors, then: (3) (i) Addition : a + b = 8 a 1 + b 1 , a 2 + b 2 < (4) (ii) Scalar multiplication: s a = 8 s a 1 , s a 2 < The definitions of vector addition and multiplication by a scalar given by (2) and (3) allow us to define subtraction of two vectors. First, we define the negative of a vector b by scalar multiplication: (5) - b ª H - 1 L b = 8H - 1 L b 1 , H - 1 L b 2 < = 8 - b 1 , - b 2 < Then we use the definition for addition of two vectors: (6) a - b = 8 a 1 , a 2 < + 8 - b 1 , - b 2 < = 8 a 1 - b 1 , a 2 - b 2 < The geometric representation of vector addition, subtraction, and scalar multiplication are illustrated in Figures 2 -4 with the two vectors a = 8 2, 2 < , b = 8 1, 4 < : Figure 2 0.5 1 1.5 2 2.5 3 3.5 x 1 2 3 4 5 6 7 y a b c Figure 3 1.5 2 2.5 x -4 -2 2 4 y a b c = a - b 2 Vector_Tensors.nb
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Figure 4 0.5 1 1.5 2 2.5 3 3.5 4 x 0.5 1 1.5 2 2.5 3 3.5 4 y a c = 1.5a Figure 3 shows that the subtraction of two vectors c = a - b can be represented as a vector with its base at the origin of the coordinate system or with its base at the terminal point of vector b The principles of ordered pairs, addition of vectors, and multiplication by a scalar defined by (3) and (4) can be used to derive the following properties of vectors in !
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Vector_Tensors - A Primer on Vectors, Basis Sets and...

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