Question

An air-traffic controller observes two aircrafts on his radar screen. The first is at altitude $800\mathrm{m}$, horizontal distance $19.2\mathrm{km}$, and $25.0°$south of west. The second aircraft is an altitude $100m$, horizontal distancelocalid="1663677399022" $17.6\mathrm{km}$, 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).

Short answer

The distance between the two aircraft is $2.29km$.

Step-by-step solution

Finding the vector from the ground under the controller of the first airplane

The position vector from ground of first plane is

$\overrightarrow{r1}=(19.2km)(\cos 25°)\hat{i}+(19.2km)(\sin 25°)\hat{j}+(0.8km)\hat{k}=(17.4\hat{i}+8.11\hat{j}+0.8\hat{k})km$

The position vector of second plane is

$\overrightarrow{r2}=(17.6km)(\cos 20°)\hat{i}+(17.6km)(\sin 20°)\hat{j}+(1.1km)\hat{k}=(16.5\hat{i}+6.02\hat{j}+1.1\hat{k})km$

Finding the displacement from the first plane to second

The displacement from the first plane to the second plane is

$\overrightarrow{r2}-\overrightarrow{r1}=(-0.863\hat{i}-2.09\hat{j}+0.3\hat{k})km$

With magnitude,

$\sqrt{{(0.863)}^2+{(2.09)}^2{(0.3)}^2}km=2.29km$