The natural way to describe the position of any point is to use
Cartesian coordinates. In two dimensions it's quite easy.
We just have. Picture like this and so we have an X
axis and Y Axis.
Origin oh where they cross and if we want to have vectors in
that arrangement, what we would have is a vector I associated
with the X axis and a vector Jay associated with the Y axis.
What all these vectors I&J?
Well, they have to be unit
vectors. A unit vector I under
unit vector. J.
In order to make sure that we do know that they are
unit vectors, we can put little hat on the top.
So if we have a point P.
And let's say the coordinates of that point are three, 4, then
the position vector of P which remember is that line segment
joining oh to pee is 3 I.
Plus four jazz.
Notice the crucial difference. That's a set of coordinates
which refers to the point that's the vector which refers
to the position vector. So point and position vector are
not the same thing.
We can write this as a column vector 34.
And sometimes. This is used sometimes that one
is used, just depends.
What about moving then into 3
dimensions? We've got XY and of course the tradition
is to use zed.
So let's have a look. Let's draw in our three axes.
So then we've got XY.
And zed.
I'll always write zed with a bar through it that so it doesn't
get mixed up with two. I don't want the letters Ed being
confused with the number 2.
So I've got these three axes or at right angles to each other
and meeting at this origin. Oh, and of course I'm going to
describe any point P by three coordinates XY and Z.
Now when I drew up this set of axes, I indicated them.
Quite easily.
I could of course Interchange
X&Y. I might choose to
interchange Y&Z. But this is the standard way. Why is it the
standard way? What is it about this that makes it the standard
way? It's standard because it's what we call a right?
Hand.
System.
Now. How can we describe workout? What is a right hand
system? Take your right hand and hold it like this.
Middle finger. Full finger and thumb at right angles.
This is the X axis.
The middle finger. This is the Y axis, the thumb.
Now rotate as though we were turning in a right handed screw.
And we rotate like that.
And so the direction in which we're moving this direction
becomes zed axis. So we rotate
from X. Why?
And we move in the direction of the Z axis. So right hander
rotation as those screwing in a screw right? Handedly notice
that it works whatever access we choose. So if we take this to be
Y again, the thumb and we take this to be zed then if we make a
right handed rotation from why route to zed, we will move along
the X axis. So let's do that.
You can see that as we rotate it, we are moving right handedly
along the X axis and you can try the same for yourself in terms
of rotating from X to zed and moving along the Y axis. So
that's our right handed system. So let's have a look at that in
terms of having a point P that's got its three coordinates XY.
And said
X.
And why?
And said now origin, oh.
Will take a point P anywhere there in space. What we're
interested in is this point P. It's got coordinates, XY and
zed. And its position vector is that line segment
OP. And so we can write down,
Oh, P. Bar is equal to XI.
Plus YJ. Plus, Zed and the unit vector
that is in the direction of the Zed Axis is taken to be K.
So again, notice the difference. These are the
coordinates XY, zed. This is the position vector
coordinates and position vector are different.
Coordinates signify appoint, position vector
signifies a line segment. We sometimes write again
as we did with two dimensions. We sometimes
write this as a column vector XY zed.
Now there are various things we would like to know and certain
notation that we want to introduce for start. What's the
magnitude of Opie bar? What's the length of OP? Well, let's
drop a perpendicular down into the XY plane there and then.
Let's join this up.
The axes there and across there.
Now let's just think what this means this length here.
Is the distance of the point above the XY plane, so it must
be of length zed.
This length, here and here is the same length. It's the
distance along the X coordinate, so that must be X.
And that's also X. Similarly, This is why and so that must be
why as well.
So if we join up from here out
to here. What we have here is a right angle triangle, and of
course we've got a right angle
triangle here as well. So this length here. There's are right
angle this length using Pythagoras must be the square
root of X squared plus Y squared, and so because we've
got a right angle here, if we use Pythagoras in this triangle
then we end up with the fact that opie, the modulus of Opie
Bar is the square root of. We've got to square that and add it to
the square of that. So that's just X squared plus.
Y squared plus Zed Square.
Now I'm going to draw this diagram again, but I'm going to
try and miss out some of the extra lines that we've added.
So XY.
Zedd.
We'll take our point P with position vector OP bar.
Again.
Drop that perpendicular down on to the XY plane.
Draw this in across here.
And that in there.
Now. This line OP makes an angle with this
axis here.
It makes an angle Alpha.
And if I draw it out so that we
can see it. Let me call this a.
If we draw out the triangle so that we can actually see
what we've got, then we've got the line.
From O to a.
There. Oh, to A and we've got this line going out here from A
to pee and that's going to be at right angles there like that.
And so if we now join P2O, we can see the angle here, Alpha.
Now we know the length of this line. We know that it is the
square root of X squared plus Y squared plus zed squared and we
also know the length of this line, it's X.
And that is a right angle, and so therefore we can write down
cause of Alpha is equal to X over square root of X squared
plus Y squared plus zed squared.
Why have we chosen this? Well, cause Alpha is what is known
as a direction.
Cosine
be cause. It is the cosine of an angle that in some way helps
to specify the direction of P.
An Alpha is the angle that Opie makes with the X axis. So of
course what we can do for the X axis we can do for the Y axis
and for the Z Axis.
So we have calls Alpha which will be X over the square
root of X squared plus Y squared plus zed squared.
Kohl's beta which will be the angle that Opie makes with
the Y axis, and so it will be why over the square root of X
squared plus Y squared plus zed squared and cause gamma.
Gamma is the angle that Opie makes with the Z Axis, and so it
will be zed over the square root of X squared plus Y squared
close zed square.
So these are our direction cosines. These are expressions
for being able to calculate them, but there is something
that we can notice about them. What happens if we square them
and add them? So what do we get if we take 'cause squared
Alpha plus cause squared beta plus cause squared gamma?
So let's just calculate this expression.
Kohl's squared Alpha is going to be X squared over
X squared plus Y squared plus said squared.
Call squared beta is going to be Y squared over X squared
plus Y squared plus zed squared.
And cost squared gamma is going to be zed squared
over X squared plus Y squared plus zed squared.
Now we're looking at adding all of these three expressions
together. Cost squared Alpha plus cost squared beta plus cost
squared gamma. Well, they've all got exactly the same denominator
X squared plus Y squared plus said squared, so we can just add
together X squared plus Y squared plus 10 squared in the
numerator. So that's X squared plus Y squared zed squared all
over X squared plus Y squared plus said squared. Of course,
that's just one.
So the squares of the direction cosines added together give us
one. What possible use could that be to us? Well, one of
the things it does mean is that we have the vector,
let's say cause.
Alpha I plus cause beta J
plus cause Gamma K.
That vector is a unit vector. It's a unit vector because if
we calculate its magnitude that's cost squared Alpha plus
cost squared beta plus cost squared gamma is equal to 1.
Take the square root. That's one. So this is a unit vector.
Further, this is X over X squared plus Y squared plus Z
squared Y over X squared plus Y squared plus said squared.
And zed over X squared plus Y squared plus said Square and so
it's in the same direction as our original OP. Our original
position vector opi bar.
And that means that this is a unit vector in the direction of
OP bar and that may prove to be quite useful later on when we
want to look at unit vectors in particular directions. For now,
let's just have a look at doing a little bit of calculation.
Let's say we've got a point.
That has coordinates 102.
Under point that has coordinates 2 - 1.
4.
The question that we might ask is if we form the vector AB.
What's the magnitude of a bee?
And what are its direction
cosines? We just have a look at this. Let's
remember that, oh, a bar.
Is.
I.
No JS.
And two K's.
That OB bar.
Will be. Two I.
Minus one J plus 4K.
We want to know what's the magnitude of the vector AB bar.
Just draw quick picture just to remind ourselves of how to get
there. There's A and its position vector with respect to.
Oh there's B with its position vector with respect to. If we're
wanting a baby that's from there to there and so we can see that
by going from A to B, we can go round AO plus OB.
And so therefore, that is OB bar minus Oh, a bar. So that's what
we need to do here. A bar must be OB bar minus oh, a bar.
And all we do to do the subtraction is what you would
do naturally, which is to subtract the respective bits
so it's two I take away I. That's just an eye bar.
Minus J takeaway no JS, so that's minus J
Bar and 4K takeaway 2K. That's plus 2K.
So now we have our vector AB bar. We can calculate its
magnitude AB modulus of a bar that's just a be the length from
A to B, and that's the square root of 1 squared plus minus one
squared +2 squared altogether. That's 1 + 1 + 4 square
root of 6, and the direction
cosines. Our cause Alpha.
That's The X coordinate over the modulus, so that's
one over Route 6.
Kohl's beta that's minus one over Route 6. the Y
coordinate over the modulus and cause gamma.
The zed coordinate over the modulus.
Now this is a fairly standard calculation. The sort of
calculation that it will be
expected. You'll be able to do and simply be able to work your
way through it very quickly. Very, very easily, so you have
to be able to practice some of these. You have to be able to
work with it very rapidly, very, very easily, but always keep
this diagram in mind.
That to get from A to B to form the vector AB bar, you go a
obarr plus Obiba and so.
It's the result, so to form a B it's Obi bar, take away OA bar.