A radar station locates a sinking ship at range 16.7 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 2.40 km.

(a) Write the displacement vector FROM plane TO ship, letting i represent east, j north, and k up.

(b) How far apart are the plane and ship?

(a)i = [19.6 * cos(153º - 90º)] -

[16.7 * cos(136º - 90º)]

j = [19.6 * sin(153º - 90º)] -
[16.7 * sin(136º - 90º)]

k = 2.40 - 0

(b) d = √(i^2 + j^2 + k^2)

To answer these questions, we can use vector calculations. Let's break it down step by step:

(a) Write the displacement vector FROM plane TO ship, letting i represent east, j represent north, and k represent up.

To find the displacement vector from the plane to the ship, we first need to determine the individual components of the displacement vectors in the east, north, and up directions.

First, let's calculate the east-west (horizontal) component:
- The range of the ship from the radar station is given as 16.7 km at a bearing of 136° clockwise from north.
- Calculate the east-west component by multiplying the range by the cosine of the bearing (136° clockwise from north):
Horizontal Component = 16.7 km * cos(136°) = -8.92 km
(Note: we use a negative value to indicate the direction is to the west)

Next, let's calculate the north-south (horizontal) component:
- The range of the plane from the radar station is given as 19.6 km at a bearing of 153° clockwise from north.
- Calculate the north-south component by multiplying the range by the sine of the bearing (153° clockwise from north):
Vertical Component = 19.6 km * sin(153°) = -17.43 km
(Note: we use a negative value to indicate the direction is to the south)

Finally, let's determine the vertical (up) component:
- The elevation of the plane is given as 2.40 km.
- Therefore, the vertical component is simply 2.40 km.

Now we can write the displacement vector from the plane to the ship:
Displacement Vector = (-8.92 km i) + (-17.43 km j) + (2.40 km k)

(b) How far apart are the plane and ship?

To calculate the distance between the plane and the ship, we can use the magnitude (length) of the displacement vector we found in part (a).

The magnitude of a vector is calculated as the square root of the sum of the squares of its components.

So, to find the distance:
Distance = √[(-8.92 km)^2 + (-17.43 km)^2 + (2.40 km)^2]

Simplifying this calculation will give you the final answer for the distance between the plane and the ship.