Vector Displacement| Vector Addition and Subtraction: Graphical Method| Two-Dimensional Kinematics| College Physics Problem 3.5

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 3.56, then this problem finds their sum R=A+B.)




First, we solve for the x and y components of vectors A and B.

For vector A, the components are:

A_x=-\left(12.0\:m\right)sin\:\left(20^{\circ} \right)=-4.1042\:m

A_y=\left(12.0\:m\right)cos\left(20^{\circ} \right)=11.2763\:m

For vector B, the components are:

B_x=-\left(20\:m\right)cos\left(40^{\circ} \right)=-15.3209\:m

B_y=-\left(20\:m\right)sin\:\left(40^{\circ} \right)=-12.8557\:m

Since we know the components, we can now solve for the resultant vector R by adding the components together. 

The resultant of the x-components is:


The resultant of the y-components is:


The magnitude of the resultant can be solved using the Pythagorean Theorem. That is:


The compass direction is

\theta =tan^{-1}\left|\frac{R_y}{R_x}\right|=tan^{-1}\left(\frac{1.5794}{19.4251}\right)=4.6483^{\circ}

The compass direction is 4.6483^{\circ} \:South\:of\:West.