第2章ベクトル

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life we naturally invoke the concept of vector components in a rectangular coordinate system. For example, if you ask someone for directions to a particular location, you will more likely be told to go 40 km east and 30 km north than 50 km in the direction $37\text{°}$ north of east.

In a rectangular (Cartesian) xy-coordinate system in a plane, a point in a plane is described by a pair of coordinates (x, y). In a similar fashion, a vector $\overrightarrow{A}$ in a plane is described by a pair of its vector coordinates. The x-coordinate of vector $\overrightarrow{A}$ is called its x-component and the y-coordinate of vector $\overrightarrow{A}$ is called its y-component. The vector x-component is a vector denoted by ${\overrightarrow{A}}_x$. The vector y-component is a vector denoted by ${\overrightarrow{A}}_y$. In the Cartesian system, the x and y vector components of a vector are the orthogonal projections, as illustrated in Figure 2.16, of this vector onto the x- and y-axes, respectively. In this way, following the parallelogram rule for vector addition, each vector on a Cartesian plane can be expressed as the vector sum of its vector components:

As illustrated in Figure 2.16, vector $\overrightarrow{A}$ is the diagonal of the rectangle where the x-component ${\overrightarrow{A}}_x$ is the side parallel to the x-axis and the y-component ${\overrightarrow{A}}_y$ is the side parallel to the y-axis. Vector component ${\overrightarrow{A}}_x$ is orthogonal to vector component ${\overrightarrow{A}}_y$.

It is customary to denote the positive direction on the x-axis by the unit vector $\widehat{i}$ and the positive direction on the y-axis by the unit vector $\widehat{j}$. Unit vectors of the axes, $\widehat{i}$ and $\widehat{j}$, define two orthogonal directions in the plane. As shown in Figure 2.16, the x- and y- components of a vector can now be written in terms of the unit vectors of the axes:

The vectors ${\overrightarrow{A}}_x$ and ${\overrightarrow{A}}_y$ defined by Equation 2.11 are the vector components of vector $\overrightarrow{A}$. The numbers $A_x$ and $A_y$ that define the vector components in Equation 2.11 are the scalar components of vector $\overrightarrow{A}$. Combining Equation 2.10 with Equation 2.11, we obtain the component form of a vector:

If we know the coordinates $\text{b}(x_b,y_b)$ of the origin point of a vector (where b stands for “beginning”) and the coordinates $\text{e}(x_e,y_e)$ of the end point of a vector (where e stands for “end”), we can obtain the scalar components of a vector simply by subtracting the origin point coordinates from the end point coordinates:

When we know the scalar components $A_x$ and $A_y$ of a vector $\overrightarrow{A}$, we can find its magnitude A and its direction angle ${\theta }_A$. The direction angle—or direction, for short—is the angle the vector forms with the positive direction on the x-axis. The angle ${\theta }_A$ is measured in the counterclockwise direction from the +x-axis to the vector (Figure 2.18). Because the lengths A, $A_x$, and $A_y$ form a right triangle, they are related by the Pythagorean theorem:

This equation works even if the scalar components of a vector are negative. The direction angle ${\theta }_A$ of a vector is defined via the tangent function of angle ${\theta }_A$ in the triangle shown in Figure 2.18:

When the vector lies either in the first quadrant or in the fourth quadrant, where component $A_x$ is positive (Figure 2.19), the angle $\theta$ in Equation 2.16 is identical to the direction angle ${\theta }_A$. For vectors in the fourth quadrant, angle $\theta$ is negative, which means that for these vectors, direction angle ${\theta }_A$ is measured clockwise from the positive x-axis. Similarly, for vectors in the second quadrant, angle $\theta$ is negative. When the vector lies in either the second or third quadrant, where component $A_x$ is negative, the direction angle is ${\theta }_A=\theta +180\text{°}$ (Figure 2.19).