Directed line segment

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.)... 

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product ... 

The expression 5/ is often pronounced 5 modulo equivalence or S mod equivalence. If possible and convenient, we refer to the equivalence by name for example, vectors are directed line segments modulo translation and antiderivatives are functions modulo constants . We leave the details of applying Definition 1.3 to vectors and antiderivatives to the interested reader in Exercises 1.19 and 1.20. 

Vectors. A physical quantity that has both magnitude and direction can be represented by a directed line segment or vector in three-dimensional space. Let A be a vector quantity. (We use boldface type for vectors.) We set up a Cartesian coordinate system xyz and denote vectors of unit length along the x, y, and z axes by i, j, and k, respectively. If Ax, Ayy and Az are the projections of A on the x, y, and z axes, then A is given by... 

A vector is represented mathematically by a directed line segment, the length of which corresponds to the magnitude of the vector, whilst... 

Figure 5.2 Representations of the vectors a, 2a and -a as directed line segments...

The vectors a and b are said to be equal if their magnitudes and directions are the same, irrespective of the locations of their respective initial points. Hence, any directed line segment with the same length and direction as a is represented by a. 

The directed line segment OQ represents the vector c, defined as the sum of a and b. Furthermore, as OQ = OP + PQ = OR + RQ, it follows that c = a + b = b + a, from which we see that addition is commutative in other words, a displacement OR followed by RQ clearly leads to Uie same final point as a displacement OP followed by PQ. 

In this system of coordinates, if a point P has the coordinates x,y,z), then the directed line segment OP, extending from the origin O to point P, corresponds to the vector r. If we apply the triangle rule twice, we obtain ... 

Figure 5.16 The area of the parallelogram OPQR is given by la X b, where the vectors a and b represent the directed line segments OR and OP. respectively...

The directed line segment beginning at O and ending at P is the position vector of the object. We denote the position vector in two dimensions by the boldface Greek letter p. In the figure, we draw an arrowhead on the directed line segment to make its direction clear. 

The location of the point P is specified by x, y, and z, which are the Cartesian coordinates of the point. These are the distances from the origin to the points on the axes reached by moving perpendicularly from P to each axis. These coordinates can be positive or negative. In the first octant, x, y,and z are all positive. In the second octant, x is negative, but y and z are positive. The point P can be denoted by its coordinates, as (x, y, z). The directed line segment from the origin to P is... 

Figure 16. A three-dimensional representation of the VSCC isosuifaees for the nonbridging oxide anion of the H6Si207 molecule. The section in the VSCC is cut parallel to the HSiO plane. The white sphere defines the position of the oxide anion and the gray one defines the position of the H atom. The directed line segments define the values of the isosmfaces (Beverly 2000).

According to classical mechanics, the state of a point-mass particle is specified by its position and its velocity. If the particle moves in three dimensions we can specify its position by the three Cartesian coordinates x, y, and z. These three coordinates are equivalent to a three-dimensional vector, which we denote by r and call the position vector. This vector is a directed line segment that reaches from the origin of coordinates to the location of the particle. We call x, y, and z the Cartesian components of the position vector. We will denote a vector by a letter in boldface type, but it can also be denoted by a letter with an arrow above it, as in T, by a letter with a wavy underscore, as in r, or by its three Cartesian components listed inside parentheses,... 

Since vectors are viewed as directed line segments, this makes it natural to view a bivector as a directed plane segment. The direction, in this case, specifies which side of the plane is up in a right-handed coordinate system, this is the same as the side that the cross product points away from. Unlike the cross product, however, the direction of a bivector is not changed by inversion in the origin. Because we have shown that = — 1, the inverse relation follows immediately ... 

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