Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 14

A vector v is a mathematical entity that possesses both a magnitude and a direction. The vector is usually a physical quantity that is independent of the coordinate system that will be used to describe its properties. A very convenient approach is first to define an orthonormal set of coordinates (x1,x2,x3). Three axes are drawn in the x1,x2 and x3 directions, which are all at right angles to one another. Such a system is often described as a Cartesian set of coordinates. The positive directions of the x1, x1 and x3 axes are described by unit vectors i1 i2 and i3, respectively, where i1=(1,0,0),i2=(0,1,0),i3=(0,0,1). The unit vectors are such that

(2.1)

The three coordinates (x1,x2,x3) describe the location of a point x that is known as the position vector, which may be written as x=x1i1+x2i2+x3i3. In tensor theory based on Cartesian coordinates, this is written in the shorter form x=xkik where a summation over values k = 1, 2, 3 is implied when a suffix is repeated (k in this example). Any vector v may be written as v=v1i1+v2i2+v3i3, or as v=vkik when using tensor notation. The scalar quantities vk, k = 1, 2, 3, are the components of the vector v with values depending on the choice of coordinates. The magnitude of the vector v is specified by

(2.2)

and its value is independent of the system of coordinates that is selected. The magnitude is, thus, an invariant of the vector.

The unit vector in the direction of the vector v is specified by v/|v|. Examples of vectors that occur in the physical world are forces, displacements, velocities and tractions.