



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 
