Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The word problems encountered most often with vectors are navigation problems. These navigation problems use variables like speed and direction to form vectors for computation. Some navigation problems ask us to find the groundspeed of an aircraft using the combined forces of the wind and the aircraft. For these problems it is important to understand the resultant of two forces and the components of force.
A really fun of application of vectors is navigation problems and by navigation problems I mean problems that involve plotting the course for an airplane. Now first of all in the airplane let's say that this is our airplane, an airplane has a velocity vector and the velocity is represented by two quantities first of all the airspeed that's the magnitude of the velocity. Airspeed, that's the speed of the airplane with respect to the air around it but there's also the direction and that's called the heading.
Now in navigation, we put this aside, the heading is measured clockwise from north and that's going to be a little different force because we usually measure counterclockwise from the positive x axis so this heading, this could be our heading of or let's say 110 and 20 degrees something like that so a heading of 120 degrees and airspeed of say 500 miles per hour, so the airspeed's the magnitude of the velocity vector, the heading is its direction.
Now the other variable in these navigation problems is wind. Wind has speed and direction and so let me draw that let's say the wind's blowing in this direction. Now the wind vectors starts at this point and goes here we're imagining that the airplane is right right at this point, I'll call the wind vector w. The magnitude of this vector is the wind speed and the wind speed maybe it's going to be 50 miles an hour let's say. The wind direction, we always measure the direction that the wind is coming from now not the direction that it's pointing to so the wind direction will actually be this angle so that's going to be wind direction and in this picture maybe it's 25 degrees.
Now when you have an airplane that's got a heading and airspeed that it's represented by this vector and you've got wind that's pointing in this direction, the wind is going to have the combined effect with the airspeed of the airplane and there's going to be a resultant of the two forces the two are the two velocities, they add up together and they get they give you a sum, so let's say that this is the sum of the two vectors and we'll call it, I don't know, g now the magnitude of g is the groundspeed this is how fast the airplane is going with respect to the ground, I'll put the airplane back here because I know it's fun. This is the groundspeed but it's also important to know what direction the plane is actually traveling with respect to the ground because that's where the plane is going so that's going to be this angle and that's called the course so just remember you know the the the plane is pointed in the direction that the heading that the heading provides. It's heading at 120 degrees and it's heading at 500 miles per hour but the wind is blowing it off course so its true course is measured by this angle the course angle and its true speed is going to be the groundspeed it will either be helped or hindered by the wind so these are terms that we're going to be using in our navigation problems usually we'll be asked to find the the true course and groundspeed given the velocity and the wind.