In Section 2.3 we added
vectors by using a scale diagram and by using properties of right triangles.
Measuring a diagram offers only very limited accuracy, and calculations with
right triangles work only when the two vectors are perpendicular. So we need a
simple but general method for adding vectors. This is called the method of
components. To define what we mean by the components of a vector A, we begin with a rectangular (Cartesian) coordinate system of
axes (Figure 1). We then draw the vector with its tail at O, the origin of
the coordinate system. We can represent any vector lying in the xy-plane
as the sum of a vector parallel to the x-axis and a vector parallel to the
y-axis. These two vectors are labeled Ax and Ay in Fig. 1.17a; they are called the component vectors of vector of A, and their vector sum
is equal to A. In symbols,
 |
| Figure 1 |
A = Ax + Ay
Since each component
vector lies along a coordinate-axis direction, we need only a single number to
describe each one. When Ax points in the positive x-direction, we define the number Ax to be equal to the magnitude of Ax. When Ax points in the negative x-direction, we define the number Ax to be equal to the negative of that magnitude
(the magnitude of a vector quantity is itself never negative). We define Ay in the same way.
 |
| Figure 2 |
We can calculate the
components of the vector A if we know its magnitude A and its direction. We’ll describe the direction of a vector by its
angle relative to some reference direction. In Fig. 1.17b this reference
direction is the positive x-axis, and the angle between vector A and the positive x-axis is (theta). Imagine that the vector A originally lies along the postive x-axis and that you then rotate
it to its correct direction, as indicated by the arrow in Fig. 2 on the angle
theta. If this rotation is from the postive x-axis toward the postive y-axis, then
the angle is postive, otherwise negative.
