Chapter 3, Problem 54AP

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.2 km, and 25.0° south of west. The second aircraft is at altitude 1 100 m, horizontal distance 17.6 km, and 20.0° south of west. What is the distance between the two aircraft? (Place the x axis west, the y axis south, and the z axis vertical.)

The position vector from the ground under the controller of the first airplane is

$\begin{array}{l}\stackrel{\to }{{\text{r}}_{\text{1}}}=(19.2\text{km)(cos25}°\text{)}\hat{\text{i}}+(19.2\text{km)(sin25}°\text{)}\hat{\text{j}}+(0.8\text{km)}\hat{\text{k}}\\ =(17.4\hat{\text{i}}+8.11\hat{\text{j}}+0.8\hat{\text{k}})\text{km}\end{array}$

The second is at

$\begin{array}{l}\stackrel{\to }{{\text{r}}_2}=(17.6\text{km)(cos20}°\text{)}\hat{\text{i}}+(17.6\text{km)(sin20}°\text{)}\hat{\text{j}}+(1.1\text{km)}\hat{\text{k}}\\ =(16.5\hat{\text{i}}+6.02\hat{\text{j}}+1.1\hat{\text{k}})\text{km}\end{array}$

Now the displacement from the first plane to the second is

$\stackrel{\to }{{\text{r}}_1}-\stackrel{\to }{{\text{r}}_2}=(-0.863\hat{\text{i}}-2.09\hat{\text{j}}+0.3\hat{\text{k}})\text{km}$

with magnitude

$\sqrt{{(0.863)}^2+{(2.09)}^2+{(0.3)}^2\text{km = 2}\text{.29}\text{km}}$