第2章ベクトル

To describe locations of points or vectors in a plane, we need two orthogonal directions. In the Cartesian coordinate system these directions are given by unit vectors $\widehat{i}$ and $\widehat{j}$ along the x-axis and the y-axis, respectively. The Cartesian coordinate system is very convenient to use in describing displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. When describing rotation, we usually work in the polar coordinate system.

In the polar coordinate system, the location of point P in a plane is given by two polar coordinates (Figure 2.20). The first polar coordinate is the radial coordinate r, which is the distance of point P from the origin. The second polar coordinate is an angle $\phi$ that the radial vector makes with some chosen direction, usually the positive x-direction. In polar coordinates, angles are measured in radians, or rads. The radial vector is attached at the origin and points away from the origin to point P. This radial direction is described by a unit radial vector $\widehat{r}$. The second unit vector $\widehat{t}$ is a vector orthogonal to the radial direction $\widehat{r}$. The positive $+\widehat{t}$ direction indicates how the angle $\phi$ changes in the counterclockwise direction. In this way, a point P that has coordinates (x, y) in the rectangular system can be described equivalently in the polar coordinate system by the two polar coordinates $(r,\phi )$. Equation 2.17 is valid for any vector, so we can use it to express the x- and y-coordinates of vector $\overrightarrow{r}$. In this way, we obtain the connection between the polar coordinates and rectangular coordinates of point P:

To specify the location of a point in space, we need three coordinates (x, y, z), where coordinates x and y specify locations in a plane, and coordinate z gives a vertical position above or below the plane. Three-dimensional space has three orthogonal directions, so we need not two but three unit vectors to define a three-dimensional coordinate system. In the Cartesian coordinate system, the first two unit vectors are the unit vector of the x-axis $\widehat{i}$ and the unit vector of the y-axis $\widehat{j}$. The third unit vector $\widehat{k}$ is the direction of the z-axis (Figure 2.21). The order in which the axes are labeled, which is the order in which the three unit vectors appear, is important because it defines the orientation of the coordinate system. The order x-y-z, which is equivalent to the order $\widehat{i}$ - $\widehat{j}$ - $\widehat{k}$, defines the standard right-handed coordinate system (positive orientation). Notice that while previously the y-axis was used for the upward direction, in three dimensions we usually choose z to represent the vertical. A right-handed coordinate system with z up and with one of the remaining axes pointing to the right and the other out of the page must have the y-axis pointing right and x out of the page, as illustrated here.

In three-dimensional space, vector $\overrightarrow{A}$ has three vector components: the x-component ${\overrightarrow{A}}_x=A_x\widehat{i}$, which is the part of vector $\overrightarrow{A}$ along the x-axis; the y-component ${\overrightarrow{A}}_y=A_y\widehat{j}$, which is the part of $\overrightarrow{A}$ along the y-axis; and the z-component ${\overrightarrow{A}}_z=A_z\widehat{k}$, which is the part of the vector along the z-axis. A vector in three-dimensional space is the vector sum of its three vector components (Figure 2.22):

If we know the coordinates of its origin $b(x_b,y_b,z_b)$ and of its end $e(x_e,y_e,z_e)$, its scalar components are obtained by taking their differences: $A_x$ and $A_y$ are given by Equation 2.13 and the z-component is given by

Magnitude A is obtained by generalizing Equation 2.15 to three dimensions:

This expression for the vector magnitude comes from applying the Pythagorean theorem twice. As seen in Figure 2.22, the diagonal in the xy-plane has length $\sqrt{A_x^2+A_y^2}$ and its square adds to the square $A_z^2$ to give $A^2$. Note that when the z-component is zero, the vector lies entirely in the xy-plane and its description is reduced to two dimensions.