Figure 1 Coordinate system

Example 10: A ship travels 75 km from port on a course of 45.0° south of east to an island. Then it travels 120 km on a course of 60.0° east of south to a second island. Find the magnitude and direction of the displacement from port.
Strategy This is just an application of vector addition using components, R_x = A_x + B_x, R_y = A_y + B_y. We denote the displacement vectors on the first and second island by A and B respectively. After finding x- and y-components for each vector, we add them to find resultant x-component and y-component. Finally, we determine the magnitude and direction of the resultant vector R using the Pythagorean theorem and the inverse tangent function.
Solution: Find the components of A.
Use Equations 2.2 and 2.3 to find the components of A.


is negative, because it points in the negative direction of y-axis.
Find the components of B:


Find the components of the resultant vector by using Eq.2.6
To find R_x add the x-components of A and B

To find R_y, add the y-components of A and B

To find the magnitude of R we use the Pythagorean theorem:

By using Eq.2.5 we can find the direction of R:

The law of cosine and sine

Figure 2.20

A and В are two vectors, and θ is the angle be­tween these two vectors as shown in Figure 2.20. The resultant vector is given as . In scalar form is

Example 11: Two vectors are shown as in the Figure 2.7. Vector A is equal to 5 and vector B is equal to 4. Angle between two vectors is 60⁰. Find the resultant vector.

We sometimes find the resultant vector without taking their components. Using the following equations and relations, Figure 2.21, we can easily find the resultant vector or one of these vectors.

Figure 2.21

The laws of sine

We use Eq.2.8 to find sides: a, b or c, Figure 2.21.

We use Eq.2.9 to find angles: A, B, C, shown in Figure 2.21.
The laws of cosine



We use Eq. 2.10 to find sides.



Eq.2.11 helps to find angles.

Figure 2.22

Example : Find unknown x-side, as shown in Figure 2.22.