Figure 1 Coordinate system

Example 9: The vector components are given by A_x = - 30 m and A_y = 40 m. Find the magnitude and direction of the vector.
Solution: We apply Equations 2.4 and 2.5 to find the magnitude and direction of the vector.
To compute the magnitude of we use the Pythagorean theorem:

Calculate the direction of using the inverse tangent.

The vector A locates at an angle of 127⁰ with respect to the positive x-axis.
Addition of two or more vectors
Figure 2.17 shows two vectors A and B. Our aim is to find the x- and y-components of the resultant vector R. We start to graph resultant vector R by connecting the initial point of vector A to the end point of vector B, as shown in Figure 2.17a. First, we will find x-component of the vector A and x-component of the vector B. We define y-components in the same way.
From Figure 2.17a we see that

Figure 2.17	Figure 2.17a	Figure 2.17b

You can see from the Figure 2.17b that the x-component R_x of the vector sum is simply the sum (A_x+ B_x=3+2=5) of the x-components of the vectors being added. The same is true for the y-components, A_y+ B_y=2+4=6.

Figure 2.17 shows this result for the case in which the components A_x A_y, B_x and B_y are all positive. Eqs. (2.6) are valid for any signs of the components of vector A and B. Let’s apply Eq. (2.6) for calculating more then, two vectors
We are given vectors A, B, and C in Fig. 2.18. The task is to find the x- and y-components of the resultant vector R. Again we start to graph resultant vector R by connecting the initial point of vector A to the end point of vector C [Fig. 2.17 (a)]. We use Eq.2.6 to find x-component and y-component of the resultant vector R. Making some calculation the resultant vector R is shown in Fig. 2.17(b).

Figure 2.18	Figure 2.18a	Figure 2.18b

Figure 2.19

If we know the components of any two vectors A and B we can compute the components of the vector sum R. Then if we need the magnitude and direction of R, we can obtain them from Eqs. (2.4) and (2.5).
A ship travels 105 km from port on a course of 55.0°west of north to an island. Then it travels 124 km due west to a second island. Then it travels 177 km on a course of 24.0° east of south to a third island.