Vectors_Tensors_03_Cartesian_Vectors

Vectors_Tensors_03_Cartesian_Vectors - Section 1.3 Solid...

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Unformatted text preview: Section 1.3 Solid Mechanics Part III Kelly 15 1.3 Cartesian Vectors So far the discussion has been in symbolic notation 1 , that is, no reference to axes or components or coordinates is made, implied or required. The vectors exist independently of any coordinate system. It turns out that much of vector (tensor) mathematics is more concise and easier to manipulate in such notation than in terms of corresponding component notations. However, there are many circumstances in which use of the component forms of vectors (and tensors) is more helpful or essential. In this section, vectors are discussed in terms of components component form . 1.3.1 The Cartesian Basis Consider three dimensional (Euclidean) space. In this space, consider the three unit vectors 3 2 1 , , e e e having the properties 1 3 3 2 2 1 = = = e e e e e e , (1.3.1) so that they are mutually perpendicular (mutually orthogonal ), and 1 3 3 2 2 1 1 = = = e e e e e e , (1.3.2) so that they are unit vectors. Such a set of orthogonal unit vectors is called an orthonormal set, Fig. 1.3.1. Note further that this orthonormal system { } 3 2 1 , , e e e is right-handed , by which is meant 3 2 1 e e e = (or 1 3 2 e e e = or 2 1 3 e e e = ). This set of vectors { } 3 2 1 , , e e e forms a basis, by which is meant that any other vector can be written as a linear combination of these vectors, i.e. in the form 3 3 2 2 1 1 e e e a a a a + + = (1.3.3) Figure 1.3.1: an orthonormal set of base vectors and Cartesian components 1 or absolute or invariant or direct or vector notation 1 e 2 e 3 e 3 3 e a a a 2 2 e a a 1 1 e a a Section 1.3 Solid Mechanics Part III Kelly 16 By repeated application of Eqn. 1.1.2 to a vector a , and using 1.3.2, the scalars in 1.3.3 can be expressed as (see Fig. 1.3.1) 3 2 2 2 1 1 , , e a e a e a = = = a a a (1.3.4) The scalars 2 1 , a a and 3 a are called the Cartesian components of a in the given basis { } 3 2 1 , , e e e . The unit vectors are called base vectors when used for this purpose. Note that it is not necessary to have three mutually orthogonal vectors, or vectors of unit size, or a right-handed system, to form a basis only that the three vectors are not co- planar. The right-handed orthonormal set is often the easiest basis to use in practice, but this is not always the case for example, when one wants to describe a body with curved boundaries (see later)....
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Vectors_Tensors_03_Cartesian_Vectors - Section 1.3 Solid...

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