3-Dimensional Vectors
3-dimensional vectors are mathematical objects that have both magnitude and direction in three-dimensional space. They are commonly represented as ordered triples of numbers and are used to describe physical quantities such as force, velocity, and displacement. In mathematics, 3-dimensional vectors are essential for understanding spatial relationships and solving problems in geometry, physics, and engineering.
7 Key excerpts on "3-Dimensional Vectors"
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...Vectors and vector spaces DOI: 10.4324/9780203829998-7 5.1 Introduction It is customary to think of vectors as entities with magnitudes and directions, and spaces as like the two-dimensional space we write in and the three-dimensional space we live in and move around in. Vectors are therefore distinct from scalars, which have magnitudes only. Our aim in this chapter is to develop a collection of results that apply to such vector entities in real n -dimensional space, or simply n -space. Our approach will be both geometric and analytic. The vector geometry will provide fresh insights into what we have already encountered in our algebraic study of n × 1 and 1 × n matrices, while the analysis will echo the matrix algebra itself. However, as we are familiar with one, two and three spatial dimensions, and can visualize more easily in these cases, we begin with vectors in 2-space (ℝ 2) and 3-space (ℝ 3) in order to fix the main ideas intuitively. It will quickly be seen that the vectors in these cases may readily be associated with 2 × 1 and 3 × 1 matrices, respectively. However, later generalization is intended not only to take us from 2- and 3-space to n -space and n × 1 matrices, but also to abstract the main properties of vectors in n -space so that they apply as well to kinds of objects other than real row or column matrices. 5.2 Vectors in 2-space and 3-space 5.2.1 Vector geometry In 2-space, also known as the plane or Euclidean plane or Cartesian plane, a simple geometric approach is to represent vectors by arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow, relative to some arbitrary datum in the plane, represents the direction of the vector. To draw a vector in the plane, we must know not only its magnitude and direction, but also its location. Thus three vectors, v, w and z, may be depicted as in Figure 5.1...
eBook - ePub- Joy Ko, Kyle Steinfeld(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
...A line, for example, can be represented by a point and a vector, as can a plane. A large class of geometric transformations may be represented by a matrix multiplied by a vector added to another vector. This naturally includes transformations, but also scalings, shearings, and skews, all of which are calculated using vectors. The geometric properties of curves and surfaces rely on the calculation of vectors, and on the classification of key sets of vectors derived from them. Given the foundational nature of vectors, ** considered along with their relative obscurity to a design audience, and their lack of visibility in CAD software, it is in line with our aim of demystifying computational geometry that this chapter opens with a thorough treatment of vectors. In the pages to follow, we will take the time to establish multiple vector representations: diagrammatically as arrows, mathematically as a tuple of numbers, and in code as an implemented class. Understanding the concepts that traverse these different representations and gaining confidence in manipulating these entities, ideally both on paper and in code, will provide a foundation on which the remainder of our survey of computational geometry will be built, and the story of the development of the Decod.es data types will be told. We begin now with a visual accounting of a vector, using the image with which we have become accustomed to associating with it: the arrow. VECTOR REPRESENTATION AND MANIPULATION A vector is an object with a length and direction, typically denoted by a letter with an arrow on top, such as v⃗. This is most often represented diagrammatically as an arrow. One caveat of this representation is that arrows with the same length and direction are considered to describe the same vector, no matter where each originates...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...3 Vector Operations As indicated previously, a vector u is defined as a quantity possessing both a magnitude and a direction; the said magnitude is regularly denoted by ‖ u ‖, while information on the direction is often conveyed graphically – or else encompasses angles formed with the axes in some reference system. Two vectors, u and v, are said to be equal when their magnitudes are identical, i.e. ‖ u ‖ = ‖ v ‖, and also point in the same direction; however, they do not need to have the same origin. A much more convenient way of handling vectors resorts, however, to their decomposition along the three directions of space in a typical Cartesian R 3 domain, according to (3.1) and (3.2) here j x, j y, and j z denote unit, orthogonal vectors of a Cartesian system, defined as (3.3) (3.4) and (3.5) – while (3.6) and (3.7) define u and v, respectively, via their coordinates. According to Pythagoras’ theorem, (3.8) and likewise (3.9) this is a more general form than Eq. (2.431), yet it relies on application of the aforementioned theorem twice. In fact, (3.10) abides to Eq. (2.431), as long as u x and u y denote the projections of u onto the x ‐ and y ‐axis, respectively, and u xy denotes the projection of u onto the x 0 y plane; further application of Eq. (2.431) then supports (3.11) where u z denotes the projection of u onto the z ‐axis. Insertion of Eq. (3.10) transforms Eq. (3.11) to (3.12) that retrieves Eq. (3.8), after taking square roots of both sides – as long as ∣ u x ∣ ≡ ‖ u x ‖, ∣ u y ∣ ≡ ‖ u y ‖, and ∣ u z ∣ ≡ ‖ u z ‖; a similar reasoning obviously applies to v x, v y, and v z describing v...
eBook - ePub- Thomas D. Wickens(Author)
- 2014(Publication Date)
- Psychology Press(Publisher)
...Chapter 2 Some vector geometry A geometric vector gives a concrete representation to a variable and to the algebraic vector of data that measures it. Before applying this geometric representation to the techniques of multivariate statistics, one needs to understand how to manipulate these vectors and combine them. The first section of this chapter describes these operations, and the second section considers the correspondence between vectors and variables. The final two sections describe the important concepts of vector spaces, linear dependence, and projection. 2.1 Elementary operations on vectors Operations such as addition and multiplication that apply to algebraic vectors have their counterparts for geometric vectors. Vectors. A vector is a directed line segment. It has two properties, its direction and its length. It can be started from any point. Both the vector that goes from the point (0, 0) to the point (2, 1) and the vector that goes from (3, 2) to (5, 3) move two units over and unit one up, and so are different instances of the same vector: The standard position from which to start a vector is at the origin 0. However, when combining several vectors or illustrating the relationships among them, it can be helpful to start them at different points. As long as neither its direction nor length is altered, one can freely slide a vector about a diagram without changing it. The length of a vector. A fundamental property of any vector is its length, which is denoted by placing vertical bars about it—the length of the vector is | |. Algebraically, the length of a vector is found by using the Pythagorean theorem. The vector = [ x 1, x 2 ]′ has length. More generally, the length of the n -dimensional vector = [ x 1, x 2, …, x n ]′ is For example, the length of the vector = [2, 1]′ in the diagram above is Scalar multiplication. The simplest operation that changes one vector into another is multiplication by a number such as 2 or − 7...
eBook - ePubLinear Optimization for Business
Theory and practical application
- Marcos Singer(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...For the example in Figure 4.2, (10, 0) = v = w − u = (16, 12) − (6, 12). Figure 4.1 Vector Representation in Three Dimensions Figure 4.2 Vector Addition Scalar multiplication consists of multiplying a vector v = (x, y, z, …) by a scalar (dimensionless) number λ, whose result is a vector λ v = (λ · x, λ · y, λ · z, …). Graphically, the arrow representing λ v is λ times the arrow representing v. If λ is negative, then λ v points to the opposite direction of v. Exercise 10 : Location on the Cartesian Plane Consider vectors a and b in Figure 4.3. The vectors p, q, …, x are in the form α a + β b. Determine in each case whether α + β is greater, equal to or less than 1, and if α and β are positive or negative values. Figure 4.3 Addition and Multiplication in the Cartesian plane Answer: The scalars α and β in the Figure meet the following conditions: p : α + β < 1; α, β > 0 q : α + β > 1; α, β > 0 r : α + β < 1; α > 0, β < 0 s : α + β > 1; α > 0, β < 0 t : α + β < 1; α < 0, β > 0 u : α + β > 1; α <. 0, β > 0 v : α + β = 1; α, β > 0 w : α + β = 1; α > 0, β < 0 x : α + β = 1; α < 0, β > 0 Vectors such as v that lie on the line between a and b are a convex linear combination of a and b. Vectors p, v and q are located inside the cone formed by a and b, while r, s, t, u, w and x are located outside the cone. A hyperspace or n-dimensional space is a mathematical entity in which points can be found. Its dimension is defined by the number of variables necessary to describe its position, i.e., by the number of Cartesian axes in the frame of reference. For example, the point x = (x, y) = (1, 2) is defined in a two-dimensional space, and the point x = (x, y, z) = (1, 2, 0) is defined in three dimensions. A one-dimensional hyperspace is represented by a line between −∞ and ∞; a two-dimensional hyperspace is represented by an infinite plane in all directions; a three-dimensional hyperspace is represented by an infinite cube in all directions; and so on...
- Michael B. Miller(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...To calculate the length of a vector, we simply take the square root of the inner product of the vector with itself: (9.5) The length of a vector is alternatively referred to as the norm, the Euclidean length, or the magnitude of the vector. Every vector exists within a vector space. A vector space is a mathematical construct consisting of a set of related vectors that obey certain axioms. For the interested reader, a more formal definition of a vector space is provided in Appendix C. In risk management we are almost always working in a space R n, which consists of all of the vectors of length n, whose elements are real numbers. SAMPLE PROBLEM Question: Given the following vectors in R 3, find the following: 1. a ⋅ b 2. b ⋅ c 3. The magnitude of c Answer: 1. 2. 3. ORTHOGONALITY We can use matrix addition and scalar multiplication to combine vectors in a linear combination. The result is a new vector in the same space. For example, in R 4, combining three vectors, v, w, and x, and three scalars, s 1, s 2, and s 3, we get y : (9.6) Rather than viewing this equation as creating y, we can read the equation in reverse, and imagine decomposing y into a linear combination of other vectors. A set of n vectors, v 1, v 2,. . ., v n, is said to be linearly independent if, and only if, given the scalars c 1, c 2,. . ., c n, the solution to the equation: (9.7) has only the trivial solution, c 1 = c 2 =. . . = c n = 0. A corollary to this definition is that if a set of vectors is linearly independent, then it is impossible to express any vector in the set as a linear combination of the other vectors in the set. SAMPLE PROBLEM Question: Given a set of linear independent vectors, S = { v 1, v 2,. . ., v n }, and a set of constants, c 1, c 2,. ...
- D.G. Roychowdhury(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...Appendix 3 Review of Vector Calculus A3.1 Introduction This appendix provides a short introduction of linear algebra, which includes vectors, matrices and tensor operations. This will help the readers revise basic concepts in linear algebra, which are important in solving conservation equations numerically using computational fluid dynamics. However, for obtaining better insight, readers are directed to books dealing with linear algebra. A3.2 Vectors and Vector Operations In fluid dynamics, we deal mostly with velocity vector v. In Cartesian coordinates, it consists of three velocity components u, v, w in x, y and z directions, respectively. It can be represented as v = u i + v j + w k where i, j, k are unit vectors in x, y, z directions as shown in Figure A.3.1. Figure A.3.1 Components of Velocity Vector v in Cartesian Coordinate System. The velocity vector v can also be represented in matrix form as shown below v = u v w The magnitude of a vector is given as v = u 2 + v 2 + w 2. Summation of two vectors v 1 and v 2 is given. as v 1 + v 2 = u 1 i + v 1 j + w 1 k + u 2 i + v 2 j + w 2 k = u 1 + u 2 i + v 1 + v 2 j + w 1 + w 2 k or v 1 + v 2 = u 1 v 1 w 1 + u 2 v 2 w 2 = u 1 + u 2 v 1 + v 2 w[--=PLGO-SE. PARATOR=--]1 + w 2. The multiplication of a scalar s with a vector v is given by s v = s u i + v j + w k = s u i + s v j + s w k. However, the multiplication of two vectors v 1 and v 2 is not so straightforward. This leads to dot product and cross product. A3.2.1 The Dot Product of Two Vectors The dot product of two vectors v 1 and v 2 is given as v 1. v 2 = v 1 v 2 c o s v 1, v 2 where c o s v 1, v 2 denotes the cosine of the angle between vectors v 1 and v 2. By. definition i. i = j. j = k. k = 1 a n d i. j = i. k = j. i = j. k = k. i = k. j = 0 Hence, v 1. v 2 = u 1 i + v 1 j + w 1 k. u 2 i + v 2 j + w 2 k = u 1 u 2 + v 1 v 2...
