You are to make four straight-line moves over a flat desert floor, starting at the origin of an xy coordinate system and ending at the xy coordinates (-140 m,30 m . The x component and y component of your moves are the following, respectively, in meters:(20 and 60) , then $\left(b_{x}and-70\right)$, then (-20 ${\mathbf{ands}}_{\mathbf{y}}$) , then (-60 and -70) . What are (a) component and (b) component ? What are (c) the magnitude and (d) the angle (relative to the positive direction of the x axis) of the overall displacement?

The vectors can be added in the same direction and equate that to the component of the resultant vector in the same direction. This would give the missing values of the components.

To evaluate the missing components, the following formulae can be used.

$$\begin{gathered} \stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}+\stackrel{\rightharpoonup }{d}=\left(a_x+b_x+c_x+d_x\right)\stackrel{\rightharpoonup }{\mathrm{i}}+\left(a_y+b_y+c_y+d_y\right)\stackrel{\rightharpoonup }{\mathrm{j}} \\ R=\sqrt{R_X^2+R_y^2} \end{gathered}$$ (i)

$$\begin{gathered} \mathrm{Given} \\ \mathrm{R}=\left(-140,30\right) \\ \mathrm{A}=\left(20,60\right) \\ \mathrm{B}=\left({\mathrm{b}}_{\mathrm{x}},-70\right) \\ \mathrm{C}=\left(-20,{\mathrm{c}}_{\mathrm{y}}\right) \\ \mathrm{D}=\left(-60,70\right) \end{gathered}$$ (ii)

To find $b_x$we have to add all x components of all vectors which is equal to-140 because components of resultant are (-140m,30m)

$$\begin{gathered} 20+b_x-20-60=-140 \\ {b}_x=-80\mathrm{m} \end{gathered}$$

Now to find $c_y$add all y components of all which is equal to 30 because components of resultant are(-140m,30m)

As components of resultant are (-140,30) , so resultant is given as

$$\begin{gathered} R=\sqrt{{\left(-140\right)}^2+{\left(30\right)}^2} \\ R=143\mathrm{m} \end{gathered}$$

$$\begin{gathered} \tan \theta =\frac{30}{-140} \\ \theta =-12° \end{gathered}$$

The angle is negative which means it is measured in clockwise direction with respect to negative x axis. This can be also concluded from the fact that the x component of resultant is negative and y component is positive. So resultant is in the second quadrant. With respect to positive x axis angle would be$180°-12°=168°$ .