Figure 1 Coordinate system

Solution: From figure 2.13(b) we can see that the magnitude of the x-component of the vector A is the length of the side adjacent to the 60⁰, angle. To find the x-component, we use Eq.2.2


Since the x-component is pointing in the positive x-direction, it must be positive. A_x = + 5 m.
It is shown in Figure 2.13b that the magnitude of the y-component of the vector A is the length of the side opposite the 60.0° angle. By using Eq.2.3 we will find the y-component,


Since the y-component is pointing in the positive y-direction, A_y = + 8.66 m.
Example 6: Now we place the vector A in the II quadrant of the Cartesian coordinate system and we will find the x- and y-components of the vector A. The vector A has a length of 10 m. It makes an angle of 155⁰ to the positive x-axis, Figure 2.14.

Figure 2.14a	Figure 2.14b

Solution: We should draw the projections of the vector A on the x-axis and on the y-axis, Fig. 2.14 (b). Then, we place an angle of α between the vector A and negative x-axis. We will find angle α as follows:
α + 155⁰ = 180⁰
α = 25⁰
Next, find the x-component as follows:

Since the x-component is pointing in the negative x-direction,

Then, find the y-component as follows:

Since the y-component is pointing in the positive y-direction,

Example 7: Determine the x- and y-components of the vector A. But in this case the vector A locates in the third quadrant of the Cartesian coordinate system. The length of the vector A is 25.0 m. It makes an angle of 255.0° to the positive x-axis as shown in Fig.2.15(a).

Figure 2.15a	Figure 2.15b

We will draw the components of the vector A on the x-axis and on the y-axis, Fig. 2.15b.

Then, we place an angle of α between the vector A and negative x-axis.
Angle α is

The magnitude of the x-component is the length of the side adjacent to angle α. Therefore, to find the x-component,


Since the x-component is pointing in the negative x-direction, A_x = - 2.58 m.
The magnitude of the y-component of the vector is the length of the side opposite angle α. Therefore, to find the y-component,

Figure 2.16

Example 8: The vector A is in IV quadrant of Cartesian coordinate system. Find the horizontal and vertical components of the vector A. The length of the vector A is 10 m and it makes an angle of 30⁰ south of east.
Solution: The triangle formed by the vector A and its components is shown in Figure 2.16. We will find the vector components of from its magnitude and direction. We use Equation 2.2 and Equation 2.3 to find the components of the vector