Locating a Sinking Ship & Rescue Plane

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A radar station locates a sinking ship at range 17.3 km and bearing 136 degrees clockwise from north. From the same station a rescue plane is at horizontal range 19.6 km, 153 degrees clockwise from north, with elevation 2.2 km.
(a) write the position vector for the ship relative to the plane (where i is east, j north and k up)
(b) how far apart are the plane and ship?


okay, first of all, i don't think i even understand the problem. I'm not sure how to draw a picture for the plane. I'm sure it'll have to be in a 3D plane, but i can't picture it. thanks.

 

[recon]

i'm sure it'll have to be in a 3D plane, but i can't picture it. thanks.

In this kind of questions, always assume the plane to be a 1D point. Picture the boat as a 1D point as well - 1D as in a dot: .

 

[plover]

You can start with a coordinate system with the radar station at the origin: the positive x-axis is east, the positive y-axis is north, and the positive z axis is up. (Thus from the radar station the vectors i, j, and k move 1 unit along, respectively, the x, y, and z axes.)

If you imagine a circle with radius 17.3 km centered on the radar station (i.e. the origin), you can locate the ship at the appropriate spot on the circle. Once you convert the angle to a more standard form, you have polar coordinates for the ship in the xy-plane – these can be converted to Cartesian coordinates. x and y coordinates can be found similarly for the plane. z coordinates for both vehicles come directly from the data you specified.

So now you have xyz-coordinates for each vehicle which are easy to rewrite as position vectors relative to the radar station. From these two vectors, it is then straightforward to find the relative position vector from one vehicle to the other.

 

[Integral]

This is a simple problem in vector addition. Draw the two vectors from the radar antenna to the plane and the ship. Now use what you have learned of Vector addition to find the vector from the plane to the ship. Think about the the triangle formed when you consider the height of the plane and the vector from the plane to the ship.