Basis Vector
Basis vectors are a set of linearly independent vectors that can be used to represent any vector in a given vector space through linear combinations. They form the building blocks for describing the geometry and physics of a space. In physics, basis vectors are often used to define coordinate systems and represent physical quantities.
5 Key excerpts on "Basis Vector"
eBook - ePub- Joy Ko, Kyle Steinfeld(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
...Notice that the same desired vector has different coordinates depending on what the basis for the space is. To make this concept more concrete, let us revisit our first encounter with local coordinates using the dot product, in which two perpendicular vectors ⃗υ 1 and ⃗υ 2 were used. Now we know that these two vectors make up a basis for a two-dimensional space. We can see that any other vector can be expressed as the sum of the projected vectors, and the local coordinates are exactly the projected lengths L 1,L 2 of the vector ⃗υ onto each of the Basis Vectors. fig 1.094 VECTOR COORDINATES IN AN ORTHONORMAL BASIS In common CAD-lingo, coordinates described relative to a non-standard basis are referred to as local coordinates, while those described relative to the standard basis of are termed world coordinates. While in CAD there is an implicit hierarchy between these two, mathematics holds these in equal esteem. Geometrically speaking, we have defined a basis as a set of vectors with the properties necessary for their use as coordinate axes. From here, a leap to a complete coordinate system, that is, something akin to the re-positionable construction plane we find in CAD, is not too far away. In fact, all that is missing is an origin, since, unlike vectors, it makes a difference where the origin of the coordinate system lies. All of the definitions laid out here, in the context of two-dimensional space, have exact counterparts in higher dimensions. Like those that span the plane, any three vectors that span the entire space of R3 is a basis for a three-dimensional space. The unique scalar multipliers c1, c2 and c3 are the coordinates of a vector in that basis. A basis together with an origin is a coordinate system in any dimension. Frames Just as two perpendicular unit vectors make up a special basis for two-dimensions, three mutually perpendicular unit vectors is a special basis in three dimensions...
- Michael B. Miller(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...Even if we can't picture it in higher dimensions, if two vectors are orthogonal, we still describe them as being at right angles, or perpendicular to each other. In many applications it is convenient to work with a basis where all the vectors in the basis are orthogonal to each other. When all of the vectors in a basis are of unit length and all are orthogonal to each other, we say that the basis is orthonormal. ROTATION In the preceding section, we saw that the following set of vectors formed an orthonormal basis for R 2 : (9.18) This basis is known as the standard basis for R 2. In general, for the space R n, the standard basis is defined as the set of vectors: (9.19) where the i th element of the i th vector is equal to one, and all other elements are zero. The standard basis for each space is an orthonormal basis. The standard bases are not the only orthonormal bases for these spaces, though. For R 2, the following is also an orthonormal basis: (9.20) SAMPLE PROBLEM Question: Prove that the following basis is orthonormal: (9.21) Answer: First, we show that the length of each vector is equal to one: (9.22) Next, we show that the two vectors are orthogonal to each other, by showing that their inner product is equal to zero: (9.23) All of the vectors are of unitary length and are orthogonal to each other; therefore, the basis is orthonormal The difference between the standard basis for R 2 and our new basis can be viewed as a rotation about the origin, as shown in Exhibit 9.5. Exhibit 9.5 Basis Rotation It is common to describe a change from one orthonormal basis to another as a rotation in higher dimensions as well. It is often convenient to form a matrix from the vectors of a basis, where each column of the matrix corresponds to a vector of the basis. If the vectors v 1, v 2,. ...
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...As any vector in ℝ 2 may be expressed in terms of v 1 and v 2,the set S = { v 1, v 2 } is a basis for ℝ 2. Figure 5.17 Change of basis using rectangular coordinates For the purposes of associating numbers, i.e. coordinates, with points in ℝ 2, given Basis Vectors v 1 and v 2, it is not essential that the basis be orthonormal. Any basis for ℝ 2 will suffice, as shown in Figure 5.18, where v 1 and v 2 are not orthonormal nor even orthogonal. Nevertheless, we can regard the numbers c and d as legitimate coordinates of the vector u, since u = c v 1 + d v 2. This generalized notion of coordinates is useful and extends to higher-dimensional Euclidean space (and more general vector spaces). Before we examine this, however, we require some preliminary results. 5.4.8.1 Preliminaries heorem 5.4.14 If S = { v 1, v 2,..., v n } is a basis for a vector space V, then every vector u ∊ V can be expressed uniquely as a linear combination of the v i, i = 1, 2,..., n, i.e. u = c 1 v 1 + c 2 v 2 +···+ c n v n in exactly one. way. Proof: Suppose u = c 1 v 1 + c 2 v 2 +···+ c n v n and u = k 1 v 1 + k 2 v 2 +···+ k n v n. Subtracting these two equations we have 0 = (c 1 – k 1) v 1 + (c 2 – k 2) v 2 +···+ (c n – k n) v n. Since v 1, v 2,..., v n are Basis Vectors, they are linearly independent and (c i – k i) = 0 for all i. Therefore c i = k i for all i. ⌖ Figure 5.18 Change of basis using non-rectangular coordinates Definition 5.4.11 (a) If S = { v 1, v 2,..., v n } is a basis for V and u = c 1 v 1 + c 2 v 2 + ··· + c n v n ∊ V, then the scalars c 1, c 2,..., c n are called the coordinates of u relative to, or with respect to, the basis S. (b) The coordinate vector of u with respect to S is the vector (u) S = (c 1, c 2,. ..., c n) ∊ ℝ n. (c) The coordinate matrix of u with respect to S is the n × 1 matrix [ u ] S = [ c 1 c 2... c n ] ┬. Example 5.4.11 Let v 1 = (3, 1) and v 2 = (–1, 2)...
eBook - ePub- Richard A. Clement(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...2.5. Cartesian and vector coordinates The procedure of adding together scalar multiples of a set of vectors is often encountered, and is referred to by the term LINEAR COMBINATION. The scalar multiples of the base vectors that are associated with a given vector are referred to as the COORDINATES of the vector, in keeping with the role of these numbers in coordinate geometry. It is also convenient to specify vectors in terms of coordinates when the result of a scalar or vector product is required. For the scalar product, the cosines of the angles between e 1 and e 1, e 2 and e 2 and e 3 and e 3 are all equal to 1, whilst the cosines of the angles between e 1 and e 2, e 2 and e 3 and e 3 and e 1 are all equal to 0, so that in terms of coordinates the definition becomes: Consideration of the vector product in terms of coordinates reveals that there are two possible directions for the e 3 vector as shown in Fig. 2.6. By convention the two different systems are referred to as a RIGHT HANDED and a LEFT HANDED system respectively. FIG. 2.6. Alternative systems of Cartesian base vectors Once it is agreed that a given system is being used, one can then choose the direction of the vector n in the cross product to be such that the following rules are satisfied: and so that in terms of coordinates: Summary A vector is a quantity which specifies both the length and the direction of a translation through space, and is denoted by boldface type. The two fundamental vector operations are the scalar product: and the vector product: CHAPTER 3 Matrix Algebra Given that the approach of this book is to follow the changes in the visual stimulus that occur along the visual pathway, the most important mathematical tool is the one used to describe these changes...
eBook - ePub- Philip C. Magnusson, Andreas Weisshaar, Vijai K. Tripathi, Gerald C. Alexander(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...A Vector Analysis–Definitions and Formulas The following presentation is meant primarily to refresh the memory of the reader in conjunction with the derivations in Chaps. 10 through 14. Rigorous treatments may be found in the references listed at the close of this appendix. The properties of vectors and their operations, and the physical concepts to which they are applied, exist independently of coordinate systems, and will be so examined initially. Coordinate systems are useful auxiliaries for detailed study of particular geometric assemblies. The forms to which the various functions of vector analysis reduce will be listed for the rectangular and circular cylindrical systems. A-1 Vector Operations and Functions Vectors are space-directed quantities; in the present text their application will be limited to functions in three-space which are descriptive of physical behavior. a. Vector Algebra 1, 3, 4 Vector-algebra operations of immediate interest are (1) the equating of one vector to another; (2) addition and subtraction of vectors; (3) scalar, or dot, multiplication of two vectors; and (4) vector, or cross, multiplication of two vectors. (1) Equality among Vectors Two vectors are said to be equal to each other if their directions are parallel and their magnitudes equal. The latter property can be true in physical applications only if the two quantities represented have the same dimensions...
