College Physics 3.5 – Resultant of two vectors


Suppose you first walk 12.0 m in a direction 20º west of north and then 20.0 m in a direction 40.0º south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.54, then this problem finds their sum R=A+B.)

The two displacements A and B are added to give a total displacement of R with magnitude R and direction θ measured from the x-axis
Figure 3.54. The two displacements A and B are added to give a total displacement of R with magnitude R and direction θ measured from the x-axis

Solution:

To solve for the magnitude and direction of the resultant R, we can refer to the triangle formed by the tree vectors as shown in the figure below.

Two vectors A and B are given and added to find for the magnitude and direction of the resultant vector R

From the figure above, we are given the two sides A and B, and the included angle measured to be 70°. To solve for the magnitude of R, we can use the cosine law.

\text{R}^2=\text{A}^2+\text{B}^2-2\text{AB}\:\cos 70^{\circ}

\text{R}=\sqrt{\text{A}^2+\text{B}^2-2\text{AB}\:\text{cos}\:70^{\circ} }\:

\text{R}=\sqrt{\text{(12 m)}^2+\text{(20 m)}^2-2\text{(12 m)(20 m)}\:\text{cos}\:70^{\circ} }\:

\text{R}=19.4892\:\text{m}

To solve for the direction of the resultant, we just need to solve for angle C, then subtract 70° to get the value of θ. We can solve for angle C using the sine law.

\displaystyle \frac{20\:\text{m}}{\sin \text{C}}=\frac{19.4892\:\text{m}}{\sin 70^{\circ} }

\displaystyle \sin \:\text{C}=\frac{20\:\text{m}\:\left(\sin \:\:70^{\circ \:}\right)}{19.4892\:\text{m}}

\displaystyle \text{C}=\text{sin}^{-1}\left(\frac{20\:\text{m}\:\left(\sin \:\:70^{\circ \:}\right)}{19.4892\:\text{m}}\right)

\text{C}=74.65^{\circ}

Finally, to solve for θ, we subtract 70° from 74.65°.

\theta =74.65^{\circ} -70^{\circ} =4.65^{\circ}

Therefore, the compass direction of the resultant displacement is 4.65° South of West.