Question

A radar station locates a sinking ship at range 17.3 km and bearing 136° clockwise from north. From the same station, a rescue plane is at horizontal range 19.6 km, 153° clockwise from north, with elevation 2.20 km.
a) Write the position vector for the ship relative to the plane, letting represent east, north, and up.
b) How far apart are the plane and ship?

Answer 1

Answer:2.2 km

Explanation:

Given

Position of ship with respect to station is $\vec{r_1}$

considering east to be positive and west to be negative x axis

North as positive y axis

$\vec{r_1}=17.3cos(-46)\hat{i}+17.3sin(-46)\hat{j}$

$\vec{r_1}=12.017\hat{i}-12.44\hat{j}$

Position vector of Plane $\vec{r_2}$

$\vec{r_2}=19.6cos(-63)\hat{i}+19.6sin(-63)\hat{j}+2.2\hat{k}$

$\vec{r_2}=12.017\hat{i}-12.44\hat{j}+2.2\hat{k}$

Position vector of ship relative to the plane

$\vec{r_{12}}= \vec{r_1}- \vec{r_2}$

$\vec{r_{12}}= 12.017\hat{i}-12.44\hat{j}-12.017\hat{i}+12.44\hat{j}-2.2\hat{k}$

$\vec{r_{12}}=-2.2\hat{k}$

Distance between Plane and Ship is 2.2

as they vertically separated by 2.2 km