Math-Prof HOME Physics Table of Contents Ask A Question PREV NEXT

Eight Miles High And Falling Fast
Vectors

 
 Much of physics is the study of motion and forces.
 Motion is something moving in some direction.
 A force is something pushing in some direction.
 Whichever one of these we talk about, 
 we have an amount AND A DIRECTION.
 The direction part is what makes a vector a vector.

 If we say "2 miles", that's just a number with some units attached.

 If we say "2 miles to the north", that's a vector.
 

 
 We are going to need a way to put these puppies into equations.
 That generally means graphing them on a set of axis.
 Sometimes (rarely) we will luck out, 
 and the vectors will all be going in the same direction.
 Most of the time, however, they will be going in inconvenient directions.
 

 
 If we just have one vector, and it is going in a straight line,
 we can just say that the direction it is going
 is right down the X axis (or the Y axis or the Q axis or whatever).
 

 
 But if we have more than one vector,
 they will probably be at crazy angles to each other.
 Lots of the time they won't wind up along an axis.
 
 Let's look at how you deal with a vector that is NOT along an axis. 
 
 Maybe this vector is part of a large system of vectors, 
 and the axis were set down to be convenient for a different vector.
 
 Anyway, start with an two axis coordinate system.
 Instead of labeling the axis X and Y, label them East and North.
 

 
 Draw a line two units long going out from the origin 
 30 degrees north of east.
 Put an arrow at the end.
 (Vectors go in some direction, if you just had the line 
 you wouldn't know which end was which.)
 

 
 When we do this for real, we will have a bunch of vectors
 and they will all be going out in strange directions.
 We will need to add up the vectors.
 This works about the same as combining like terms in Algebra.
 The deal is, if all of the vectors are shooting out in crazy directions,
 it will be hard to combine them as they sit.
 
 The way we deal with this is to translate each vector, 
 into the amount of the vector that is along each axis.
 The vector we drew above, goes mostly to the east, 
 and a little to the north.
 Instead of one vector two units long going off at an angle,
 we will wind up with two vectors.
 One that is a bit less than 2 units and goes east.
 One that is pretty short and goes to the north.
 
 If you drew a line straight down from the end of the vector to the X axis,
 and then back to the origin from there, you would have a right triangle!
 And where there are right triangles, can trig be far away?
 

 
 So here we are back at right triangle trig.
 The side opposite the angle is the length in the north direction.
 The side adjacent to the angle is the length in the east direction.
 30 degrees is our angle.
 Our vector is the hypotenuse.
 

 

 
 In the vector world, we have ...
 

 
 Remember that with trig problems, 
 if you are missing some info that you need,
 but you know some of the sides and angles, 
 you might be able to find the parts you need.
 And don't forget old Pythagoras. 
 He can come in real handy with these problems too.
 
 Example:
 
 An oil tanker travels 100 miles at a heading of 20 degrees.
 There are no currents or wind to affect the path traveled.
 How far north did it travel?
 How far east did it travel?
 
 Draw the axis, and then draw the vector.
 

 
 We need these two lengths ...
 

 
 Before we start, we need one little trick from geometry.
 The trick is called alternate interior angles.
 The axis running north and south, and the north distance line are parallel.
 The path of the tanker makes a 20 degree angle with the north south axis.
 So the path of the tanker and the north distance line
 also make a 20 degree angle.
 
 The distance north line is the side adjacent, so
 
 Cosine = adjacent (north) divided by hypotenuse
 which means adjacent = hypotenuse times cosine
 

 distance north = 100 miles cos 20

 distance north = 93.97 miles

 
 We could use another trig function to get the distance
 that the tanker moved to the east,
 but we can also use Pythagoras ...
 

 
 So the tanker traveled 93.97 miles north and 34.21 miles east.
 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Physics Table of Contents Ask A Question PREV NEXT