8/23/2005 Base Vectors.doc 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS Base Vectors A: It is very important that you understand that coordinates only allow us to specify position in 3-D space. They cannot be used to specify direction! The most convenient way for us to specify the direction of a vector quantity is by using a well-defined orthornormal set of vectors known as base vectors. Recall that an orthonormal set of vectors, say 123ˆˆˆ,,aaa, have the following properties: 1. Each vector is a unit vector: ˆˆ ˆˆ ˆˆ11 22 331aa aa aa⋅= ⋅ = ⋅ = 2. Each vector is mutually orthogonal: ˆˆ ˆˆ ˆˆ12 23 310aa aa aa⋅=⋅=⋅= Q: You said earlier that vector quantities (either discrete or field) have both and magnitude and direction. But how do we specify direction in 3-D space? Do we use coordinate values (e.g., x, y, z )??8/23/2005 Base Vectors.doc 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS Additionally, a set of base vectors 123ˆˆˆ,,aaa must be arranged such that: An orthonormal set with this property is known as a right-handed system. All base vectors 123ˆˆˆ,,aaa must form a right-handed, orthonormal set. Recall that we use unit vectors to define direction. Thus, a set of base vectors defines three distinct directions in our 3-D space! 12 3 23 1 31 2ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ,,aa a aa a aa a×=×=×= 1ˆa 3ˆa 2ˆa Q: But, what three directions do we use?? I remember that you said there are an infinite number of possible orientations of anorthonormal set!!8/23/2005 Base Vectors.doc 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS A: We will define several systematic, mathematically precise methods for defining the orientation of base vectors. Generally speaking, we will find that the orientation of these base vectors will not be fixed, but will in fact vary with position in space (i.e., as a function of coordinate values)! Essentially, we will define at each and every point in space a different set of basis vectors, which can be used to uniquely define the direction of any vector quantity at that point! A: We will in fact study one method for defining base vectors that does in fact result in an othonormal set whose orientation is fixed—the same at all points in space (Cartesian base vectors). However, we will study two other methods where the orientation of base vectors is different at all points in space (spherical and cylindrical base vectors). We use these two methods to define base vectors because for many physical problems, it is actually easier and wiser to do so! Q: Good golly! Defining a different set of base vectors for every point in space just seems dad-gum confusing. Why can’t we just fix a set of base vectors such that their orientation is the same at all points in space?8/23/2005 Base Vectors.doc 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS For example, consider how we define direction on Earth: North/South, East/West, Up/Down. Each of these directions can be represented by a unit vector, and the three unit vectors together form a set of base vectors. Think about, however, how these base vectors are oriented! Since we live on the surface of a sphere (i.e., the Earth), it makes sense for us to orient the base vectors with respect to the spherical surface. What this means, of course, is that each location on the Earth will orient its “base vectors” differently. This orientation is thus different for every point on Earth—a method that makes perfect