When we talk about forces, we often will talk about the projection of a force onto a line
or a component of a force.
It's important that you guys understand what the difference is between projections and components.
So if we start with components, two forces are components of a third if you can add them with vector addition.
So F1 + F2 = this third one.
In general, what we usually do is [when] we're working with two different axes
like the x and y axes. So if you take the unit vectors along those axes,
you can have some magnitude times one unit vector
plus some magnitude times the second unit vector
to give you the original force R.
Now in our x-y coordinate system, which you're very familiar with at this point, if I have a unit vector
-- we usually call it i along the x axis and the unit vector j along the y axis --
then I can count off on the axes and say that R is 3i - 4j.
That gives me this vector right here.
And we talk about these triangles all the time
where with any value of theta you can say F_x is R*cos(theta)
like that and find the similar one for F_y.
Now what happens if you're not dealing with axes that are perpendicular to each other?
I mean after all, the world if you look around is hardly completely rectilinear.
We'll have slanted things and posts all the time.
Probably even in your classroom you can find something where the floor is not perpendicular
to the railing or anywhere else.
In that kind of sense, what we usually use is a coordinate system along one of the axes
and another one.
But that means they're not perpendicular to each other.
The idea of the components still works though.
And these are called general force components.
I can still measure some amount up the v axis.
In this case 4 -- 1, 2, 3, 4.
And some of the out up the u axis.
In this case, 1,2,3.
And I can say that R is 3 times the unit vector along u plus 4 times the unit vector along v.
Those are the components along the u and v directions.
Now the problem is this triangle is not a right triangle.
So when you are doing this you cannot say F_v is R*cos(theta), because it's not.
You have to use the law of sines and cosines to actually find these components.
So that's a component. What's a projection?
A projection onto a line literally says drop the perpendicular from the tip of the vector
back down to the line.
So if you have this line, you're going to make this a dropped perpendicular to get here,
this length is the projection onto the line.
By definition, that's always going to be a right triangle.
So you're always going to have that the perpendicular -- the projection on the line is the
magnitude of F times the cosine of the angle in between them.
Now if you take lambda_L as the unit vector along this line like we were just doing before
-- that's the unit vector in the direction -- then what you can say is this is actually
the dot product between the vector itself and the unit vector along the line.
And if you remember what the definition of a dot product is, any time you did this in early calculus,
you get A dot B is the magnitude of A times the magnitude of B times the cosine of the angle in between them.
That's what we're sort of using here, except B is a unit vector so that the magnitude of
B is 1.
That gives you this sort of notion.
That's a projection onto a line.
So when is the projection and a component the same?
People will sometimes use them interchangeably, because when you're talking about a Cartesian
coordinate system -- which is our most common coordinate system -- they are the same.
Because these are your right triangles, the projection that you would get if you dropped
the perpendicular to the axis and the components that you get are the same.
So I will have F_x*i + F_y*j = R.
But F_x is in fact ... the component is equal to the projection.
Now if you have non-orthogonal axes, that's just not true.
So these red lines right here are the distances that you get if you drop perpendiculars to
those axes.
So this triangle right here and that triangle, those are right triangles.
So the red lines are my projection.
But the green lines have to be taken a long lines parallel to the axes.
That's easy over here because everything that's up and down, right and left is parallel to
the axes.
Here they're not.
So to have components I have to go along a line parallel to the u axis and along a line
parallel to the v axis to get out to my point.
So F_u and F_v are my components. These little red lines are my projections.
And they're not the same.
So when are projections and components the same? When your axes systems are orthogonal.