MSMS Vectors and Matrices Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2015-2016 B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 1 / 39
Introduction Most of the topics introduced in this course require the knowledge of few basic mathematical concepts and tools, namely those of VECTORS and MATRICES . Loosely speaking, vectors are used to represent many different quantities (physical and geometrical) in the three-dimensional (3D) space, and matrices are used as operators, acting on vectors. Vectors are a way to represent points in the 3D space or physical quantities that have both a magnitude and a direction. Vector may also have a number of other meanings depending on context. The 3D space is also called Euclidean Space , since we assume that it is endowed with a number of properties, that we will specify later on, coming from the axioms of geometry due to Euclid. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 2 / 39
Introduction Vectors are mainly used in this course to represent two types of mathematical or physical entities, namely geometrical quantities , like points, lines, planes, etc., and physical quantities , like velocities, accelerations, forces, torques, gradients, etc. To use vectors and operate on them, it is necessary to understand their mathematical representation. This representation may assume different forms, but in a 3D space it always consists of three real numbers, called the vector components . Vectors obey to a number of rules that will be specified later on. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 3 / 39
Introduction Matrices are mainly used in this course to represent two types of “operators”, namely rigid motions in 3D, and operators acting on vectors to transform them for some specified scope. To use matrices and operate on them, it is necessary to give them a mathematical representation. This representation may assume different forms, but in a 3D space is always determined by a row-column array of real numbers, called the matrix components . Matrices obey to a number of rules that will be specified later on. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 4 / 39
Introduction In these slides a generic vector v is written as a small boldface fonts, but often you may find other graphical symbols, small or capital, boldface or not, with arrows or underlined; see some examples − → − → ⇀ ⇀ v , v , v , V , v , v , V , v , v On the blackboard, for practical reasons, vectors will be always small underlined fonts, as in v . In these slides a generic matrix M is written in capital boldface font, but you can find the same variety of representations as with vectors. On the blackboard, for practical reasons, matrices will be always capital non-underlined fonts, as in M . Many textbooks, coming from the mechanical engineering community, represents vectors and matrices in the following way vector → { V } , matrix → [ M ] B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 5 / 39
1. Geometrical vectors The position of a geometrical point P in n -dim space is always given by n coordinates, relative to some pre-defined reference frame . The most used reference frame is the orthogonal cartesian reference frame (or simply cartesian frame). In orbit dynamics, the 3D polar frame is often used. If we want to represent a geometrical point, we use a geometrical vector p to represents it. If the point P is in the 3D space R 3 then the representation is p x p 1 p a ∈ R 3 = p y ≡ p 2 p z p 3 a a p i is the i -th coordinate; the index a indicates the reference frame R a that we use to represented the point. If we change the reference frame, the representation changes too. We will see later how to transform the point representation from one frame to another. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 6 / 39
Affine geometry To treat points as vectors implies the definition of a zero point (the origin of the reference frame), i.e., a point with particular privileged place in space. Since, in many applications, this may not be required, a particular geometry that is “origin-free” must be considered. This geometry is called affine geometry and is defined on affine spaces. Affine geometry is at the base of projective geometry and perspective transforms, as well as homogeneous vectors, that are the mathematical tools for image representation and analysis in computer graphics. Affine geometry will not be considered in this course. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 7 / 39
2. Physical vectors Many physical quantities possess both magnitude and direction. If we want to represent them, we use again the vector notation. A physical vector QP represents a physical quantity, for instance, linear or angular velocity, gravitational acceleration, force, torque, gradient, etc. Therefore, a physical vectors may be represented by an oriented segment (or directed line segment), with an application point Q (that can be free to move in space or constrained to some body), a direction and a magnitude. QP may be reference-free (e.g., physically independent from the way we represent it, or from an origin), and sometimes it is customary to represent it by the difference of two geometrical vectors (for example to represent a translation) p 1 − q 1 v 1 v QP ∈ R 3 = p − q = ≡ QP = ( P − Q ) p 2 − q 2 v 2 p 3 − q 3 v 3 B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 8 / 39
Examples Ex1: the local gravitational acceleration has direction and magnitude that are “absolute”, since they do not depend on the reference frame chosen, but on the space-time geometry. Usually, having define a local reference frame with the z -axis pointing upward, the gravity acceleration is represented by the vector 0 g = 0 − G where G varies from one place to the other, but is approximately equal to G = 9 , 81 ms − 2 on the Earth surface. Ex2: the velocity of a body is given by a vector that may have different representations in different reference frames, but its magnitude is independent from the reference frame chosen. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 9 / 39
Physical vectors The lines below give some more ideas. Figure: Physical vectors. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 10 / 39
Physical vectors We represents physical vectors using an icon; the most used icon is the arrow . Figure: The arrow icon. Obviously this arrow does not exist in space, and can often be misleading, considering the following properties of physical vectors. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 11 / 39
Polar vectors There are two types of physical vectors: polar and axial ones. Polar vectors (see [01]) are physical vectors that are symmetrical wrt a reflection through a parallel plane, and are skew-symmetrical wrt a reflection through a perpendicular plane. Examples of physical polar vectors are displacements, linear velocities and forces. Figure: Polar vector. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 12 / 39
Axial vectors Axial vectors are physical vectors that are skew-symmetrical wrt a reflection through a parallel plane, and are symmetrical wrt a reflection through a perpendicular plane. Examples of physical axial vectors are angular velocities, torques, magnetic field due to electrical currents, etc. Figure: Axial vector. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 13 / 39
Discussion The different symmetry properties of polar and axial vectors show that the arrow icon may be a misleading icon. One must therefore always keep in mind the meaning of the associated entity, that can be a geometrical point, a translation, a velocity, a torque, etc. B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 14 / 39
3. Mathematical vectors The name “vector” is also given to mathematical entities. In particular a (mathematical) vector is an abstract entity belonging to a vector space. Vector Space Given a field F = { F ;+ , ·} , a vector space V ( F ), is the set of elements, called vectors , and two operators + and · , that satisfy the following axiomatic properties: Vector sum : the operation +, called vector sum , is defined so that { V ( F );+ } is a commutative (abelian) group; the identity element is called 0 ; v + 0 = v Product by a scalar : For each α ∈ F and each v ∈ V ( F ), it exists a vector α v ∈ V ( F ); B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 15 / 39
Vector space For each α , β ∈ F and each v , w ∈ V ( F ) the following relations hold true: associative property wrt product by a scalar: α ( β v ) = ( αβ ) v existence of the identity wrt product by a scalar: 1( v ) = v ; ∀ v distributive property wrt vector sum: α ( v + w ) = α v + α w distributive property wrt product by a scalar: ( α + β ) v = α v + β v B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 16 / 39
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