At a given instant the position of a plane at A and a train at B are measured relative to a radar antenna at O. Determine the distance d between A and B at this instant. To solve the problem, formulate a position vector, directed from A to B, and then determine its magnitude.

#### Solution:

Show me the final answer↓

We will first find out the locations of point A and point B and write them in Cartesian vector form. To do so, we will use trigonometry.

Notice how we can create a right angle triangle as highlighted in the image. Using sine and cosine functions, we can figure out the z-component of point A (the height) and F’. F’ is the force that lies on the xy plane.

F'\,=\,5\cos(60^0)\,=\,2.5 km

(Remember, sin is opposite over hypotenuse, and cosine is adjacent over hypotenuse. In this triangle, the hypotenuse is 5)

Next, we will figure out the x and y components of point A.

In this new triangle we highlighted, notice how F’ is now the hypotenuse. We will use F’ and trigonometry to figure out the x and y components.

y-component = 2.5\sin(35^0)\,=\,1.43 km

(remember we found that F’=2.5)

We can now write point A in Cartesian vector form.

(Note the negative signs in front of the i(x-term) and the j(y-term). This is because from the diagram, we see that point A lies on the negative quadrant of the x and y axes)

We will now focus on point B.

Again, we will use trigonometry to figure out the new F’ in this triangle, and the z-component (height).

F'\,=\,2\cos(25^0)\,=\,1.81 km

We will now figure out the x and y-components.

As before, note how F’ is now the hypotenuse of the triangle we just formed.

y-component = 1.81\cos(40^0)\,=\,1.39 km

(remember we found that F’=1.81)

Let us write point B in Cartesian vector form.

(Note the negative signs in front of the k(z-term). Again, this is because from the diagram, we see that point B lies below the xy plane. In other words, it is in the negative z-axis.)

We can now figure out the position vector, r_{AB}.

r_{AB}\,=\,\left\{3.21i+2.82j-5.17k\right\} km

The distance between the two points is equal to the magnitude of the position vector r_{AB}.

magnitude of r_{AB}\,=\,6.71 km

#### Final Answer: