Components of a vector are vectors, which when added, yield the vector. For example, as shown in the previous section (Figure 10), traveling 3 miles north and then 4 miles east yields a resultant displacement of 5 miles, 37 north of east. This example demonstrates that component vectors of any two non-parallel directions can be obtained for any resultant vector in the same plane. For the purposes of this manual, we restrict our discussions to two dimensional space. The student should realize that vectors can and do exist in three dimensional space.

One could write an alternate problem: "If I am 5 miles from where I started northeast along a line 37 N of east, how far north and how far east am I from my original position?" Drawing this on a scale drawing, the vector components in the east and north directions can be measured to be about 4 miles east and 3 miles north. These two vectors are the components of the resultant vector of 5 miles, 37 north of east.

Component vectors can be determined by plotting them on a rectangular coordinate system. For example, a resultant vector of 5 units at 53 can be broken down into its respective x and y magnitudes. The x value of 3 and the y value of 4 can be determined using trigonometry or graphically. Their magnitudes and position can be expressed by one of several conventions including: (3,4), (x=3, y=4), (3 at 0, 4 at 90), and (5 at 53). In the first expression, the first term is the x-component (F_X), and the second term is the y-component (F_y) of the associated resultant vector.

As in the previous example, if only the resultant is given, instead of component coordinates, one can determine the vector components as illustrated in Figure 11. First, plot the resultant on rectangular coordinates and then project the vector coordinates to the axis. The length along the x-axis is F_X,and the length along the y-axis is F_y.This method is demonstrated in the following example.

Figure 11 Vector Components First, project a perpendicular line

For the resultant vector shown in Figure 12, determine the component vectors given F_R = 50 lbf at 53. First project a perpendicular line from the head of F_R to the x-axis and a similar line to the y-axis. Where the projected lines meet, the axes determine the magnitude size of the component vectors. In this example, the component vectors are 301bf at 0 (F_x) and 401bf at 90 (F_y). If F_R had not already been drawn, the first step would have been to draw the vector.

Figure 12 Component Vectors

As an exercise, the student should graphically find the easterly and northerly components of a 13 mile displacement at 22.6 north of east. The correct answer is 5 miles east and 12 miles north.

Trigonometry may also be used to determine vector components. Before explaining this method, it may be helpful to review the fundamental trigonometric functions. Recall that trigonometry is a branch of mathematics that deals with the relationships between angles and the length of the sides of triangles. The relationship between an acute angle of a right triangle, shown in Figure 13, and its sides is given by three ratios.

Figure 13 Right Triangle

Before attempting to calculate vector components, first make a rough sketch that shows the approximate location of the resultant vector in an x-y coordinate system. It is helpful to form a visual picture before selecting the correct trigonometric function to be used. Consider the example of Figure 12, that was used previously. This time the component vectors will be calculated.

Example 1: Determine the component vectors, F_x and F_Y,for F_R = 501bf at 53 in Figure 14. Use trigonometric functions.

Figure 14 F_R = 501bf at 53

F_Y is calculated as follows:

Therefore, the components for F_R are F_x = 301bf at 0 and F_Y = 401bf at 90'. Note that this result is identical to the result obtained using the graphic method.