EXAMPLE 4.2.2: CROSS PRODUCT: SHORTEST DISTANCE

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 shortest distance the train can interfere the signal of the antenna to the plane. (Shortest distance of point B from OA).

Start by identifying the coordinates of the points A, B, and O. Being the point of origin, point O coordinates can be easily identified (0, 0, 0). For the other points A and B, the components would have to be broken down as per given. Because angles were used to designate the coordinates, trigonometric function procedure is required.

For the plane at Pt. A:

The plane's configuration can be broken down firstly between the z axis and the plane xy.

                            

                           

While Fz is already the component of the plane on the z axis, the components at the x- and y- axes should still be broken down from Fxy.

                         

                        

Both x- and y- components are negative ( going into the plane yz-  and going to the left)

For the train at Pt. B:

Start off with the plane xy and axis z just the same for the train.

                          

                          

Further break down Fxy into individual components:

                        

                       

So, coordinates of points A, B, and O are as follows:

Point	i, km	j, km	k, km
A	-2.05	-1.43	4.33
B	1.16	1.39	-0.85
O	0.00	0.00	0.00

For the shortest distance of point B from line OA, take the perpendicular distance d.

From two forces OA and OB, taking the angle  between them, we can identify the perpendicular distance, d, trigonometrically as:

                  








Also recall the geometric definition of cross product as:
                     
                                       

Since, we are solving for the distance, which is just a magnitude we can modify the cross product formula as:

                                       

We introduce the distance d into the cross product formula using the trigonometric definition:

                                     

Where our working formula for the shortest distance (perpendicular distance) between point B and line OA is:

                                      

Using this formula, we no longer need to find the value for .



For Force OA:



The scalar value for Force OA is already given as 5km. You can check this by using Pythagorean formula with the values for i-, j-, and k-.

                                  

For Force OB:



No need to further determine the scalar of force OB.


Cross product of Forces OA and OB:












Take the magnitude of the cross product




From the derived working equation:

                                   ANSWER


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