Question

A person undergoes three displacements from point i to point c as indicated in the figure below. What are the magnitude and angle of the total displacement from point i?
d1 =6.00m,d2 =8.00m,d3 =5.00m,θ1 =40°,θ2 =30°,andθ3 =0°

Answer 1

Draw the figure to understand how to solve the problem. The blue vector below represents the total displacement:

As we see, the blue displacement vector can be represented as a sum of parts of the other vectors. Calculate its distance along the horizontal:

$$d_x=d_1\text{ cos}\theta_1+d_2\text{ cos}\theta_2+d_3\text{ cos}\theta_3,\\ d_x=16.5\text{ m}.$$

The distance of the blue vector along the vertical:

$$d_y=d_1\text{ sin}\theta_1+d_2\text{ sin}\theta_2+d_3\text{ sin}\theta_3,\\ d_y=7.86\text{ m}.$$

The magnitude (length) of the blue displacement vector:

$$d=\sqrt{d_x^2+d_y^2}=18.3\text{ m}.$$

The angle of this vector is

$$\theta=\text{arctan}\frac{d_y}{d_x}=23.2°.$$