A plane flies from A to B on a bearing of N75E for 810 miles.Then it flies from point B to point C on a bearing of N32E for 648 miles.Find the distance from A

Question

A plane flies from A to B on a bearing of N75E for 810 miles.Then it flies from point B to point C on a bearing of N32E for 648 miles.Find the distance from A to C.

Answer 1

To show this most directly, I would like to be able to draw a couple vectors.  Not being able to do that, I could just say that I think the easiest way to do this is to treat each displacement as a vector and find its components.

 

With the definition of directional bearing, the angles given are such that the x-components (east-west) of each leg are related to the overall displacement of that leg through sine -- because they are opposite the given angle -- and the y-components (north-south) are related through cosine, since they are adjacent to the given angle.

 

Also, because both displacements are into what on a standard Cartesian x-y plot would be the first quadrant, I know that all the components are in what I would define to be the positive direction (east for the x components, north for the y components)

 

Thus, I would write:

 

AB_x = 810sin(75) ~ 782.4 miles

AB_y = 810cos(75) ~ 209.6 miles

 

BC_x = 648sin(32) ~ 343.4 miles

BC_y = 648cos(32) ~ 549.5 miles

 

Then the components of the overall displacement (AC) are:

 

AC_x = ABx+BCx = 782.4+343.4 = 1165.8 miles

AC_y = AB_y+BC_y = 209.6+549.5 = 759.1 miles

 

These constitute the legs of a right triangle, for which the overall magnitude of displacement (distance from Point A to Point C) is the hypotenuse.  Use the Pythagorean theorem to find the length of that hypotenuse, and you have your answer.

 

I hope this helps.  It would be much clearer if I could make a quick drawing.