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Vectors_Tensors_01_Vector_Algebra - Section 1.1 1.1 Vector...

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Section 1.1 Solid Mechanics Part III Kelly 3 1.1 Vector Algebra 1.1.1 Scalars A physical quantity which is completely described by a single real number is called a scalar . Physically, it is something which has a magnitude, and is completely described by this magnitude. Examples are temperature, density and mass . In the following, lowercase (usually Greek) letters, e.g. γ β α , , , will be used to represent scalars. 1.1.2 Vectors The concept of the vector is used to describe physical quantities which have both a magnitude and a direction associated with them. Examples are force , velocity , displacement and acceleration . Geometrically, a vector is represented by an arrow; the arrow defines the direction of the vector and the magnitude of the vector is represented by the length of the arrow, Fig. 1.1.1a. Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a , r , q . The magnitude (or length ) of a vector is denoted by a or a . It is a scalar and must be non-negative. Any vector whose length is 1 is called a unit vector ; unit vectors will usually be denoted by e . Figure 1.1.1: (a) a vector; (b) addition of vectors 1.1.3 Vector Algebra The operations of addition, subtraction and multiplication familiar in the algebra of numbers (or scalars) can be extended to an algebra of vectors. a b c (a) (b)
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Section 1.1 Solid Mechanics Part III Kelly 4 The following definitions and properties fundamentally define the vector: 1. Sum of Vectors: The addition of vectors a and b is a vector c formed by placing the initial point of b on the terminal point of a and then joining the initial point of a to the terminal point of b . The sum is written b a c + = . This definition is called the parallelogram law for vector addition because, in a geometrical interpretation of vector addition, c is the diagonal of a parallelogram formed by the two vectors a and b , Fig. 1.1.1b. The following properties hold for vector addition: a + b = b + a … commutative law a +( b + c ) = ( a + b ) + c … associative law 2. The Negative Vector: For each vector a there exists a negative vector . This vector has direction opposite to that of vector a but has the same magnitude; it is denoted by a . A geometrical interpretation of the negative vector is shown in Fig. 1.1.2a. 3. Subtraction of Vectors and the Zero Vector: The subtraction of two vectors a and b is defined by ) ( b a b a + = , Fig.
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