So from your origin of a year, we define or a which is our position vector for A as this vector or
B as that vector and then may have a relative position.
So what is the position of a relative to be that would be this or a relative to be?
And if you swapped that arrow around and you put it aside, it would be the position of B relative to
a..
Now, the nice thing is we can.
We can handle this as vector addition, so we've got these vectors and we can add them, we can say
this vector is that vector, plus that vector.
So meaning we can say that our position of eight is or B plus or A relative to be sort of Duniya, as
I've said, that one is that one plus that one just the way you would normally add vector.
OK, and we can therefore also say that if you differentiate this, we can say the early is the B plus
the A relative to B, and these are all vectors.
That's why they've got a little stripe, you know, so they're going to have iron J components because
we just work in a two dimensional case.
If we got in a three dimensional case, it would have had I, j and K components that we just going
to have Einziger components adding these vectors together.
And the same for acceleration.
Acceleration of particle A is the acceleration of particle B plus the acceleration of particle A relative
to B, and we'll do an example of that to show you how it works.
So let's consider this example.
I've done my best to try to do it properly.
But what you have here is imagine two boats.
There's Boat B in this boat and watching them from the top.
This is a shoreline.
So let's you have a beach there.
This is a sea or a lake.
And this boat is travelling in this direction.
But B.
And this boat, but I was traveling in that direction, it has a velocity of 10 meters per second and
but because of velocity of 15 meters per second.
Now note that these velocities are scalars.
In this case, they they don't include the direction yet that just in this direction in which the boat
moves, just going in that direction indicated by that arrow.
Same with this one.
So we need to break up this velocity into its components.
And luckily we have a angle here and we have an angle there to do it so we can break it down into a
component in the other direction and a component in the direction.
And then we can add those components because what we want to know is what is the velocity of about a
relative to both be OK?
And we know that that would be we know that this has a certain formula and that that would be the miners
will be on.
So that would be the A minus the B, but these all vectors.
So we first need to break this down into different components.
So let's have a look.
So what I've done is I've drawn a vector diagram of the movement of these votes like this.
So, yeah, you have and they have to be this victor of a year indicates the velocity of Bird B and
this one the velocity of but A, you can see that this vector is longer than this one because we have
velocity of 15 meters per second versus a velocity of 10 meters per second.
So this one is longer because the length indicates the magnitude of the velocity and then of course,
the arrow indicates the direction.
OK, so what we want to know is V of a relative to be so we need to know this vector of a year so we
can get this vector of a year by just adding vectors.
Or what we could do is we can use the cosine route because we just want the length of this and we can
get the angle also with a room.
So what the Cassandre says is that this length squared of the year is this is this length squared,
plus battling squared minus two, this one times that one cosine of the angle between the two.
So what it basically says is that the a relative to be so that's one of the squared
will be this one squared plus that one squared.
So we have that already.
I was going to say ten squared, plus fifteen squared, minus two this one times that one.
So it's going to be ten multiplied by 15.
OK, and then we have the cosine of this angle between the two.
So we know that angle is 30.
This is 45.
This one is 90.
So that must be 45.
OK, so this whole angle is seventy five.
Seventy five.
And so if we calculate this and we take the square root of that, we did the a relative to be.
Is equal to 15, comma seven.
Meters per second.
So that is the length of this vector we don't yet know the direction.
So let us find the direction if we define the direction of this vector with regards to the horizontal.
We basically want that angle there.
So let's call that angle up.
That'll define this direction for us from the horizontal.
But to get that, we need to find this angle here and we can call it fine, and that we can do with
a signed route.
And we know that the Sindel says that the son of this divided by this equals the sign of this angle
divided by that length of there.
So what I can say is that the sign of this angle, which you're interested in science over this length,
then as equal to the sign of this angle of a year, which is 75 signs.
Seventy five over this length of there, which is the one with the we just calculated fifteen point
seven.
OK, and now I could just solve for sign find and I can take the inverse of that and I can get that
five our angle there, that angle of the there is thirty seven point eight, nine degrees.
OK, now I want that angle between that vector, the A, B and the horizontal.
So we just need to subtract our angles from each other and then we can find that relative angle.
So what we know is that this is 45 and this is the horizontal it's side and that is also horizontal.
So if you go like this, this whole angle of the year must be 45 as well.
Right.
So we know that that angle there is just going to be 45 degrees, minus five.
So Seeta is 45 degrees, minus thirty seven, eight, nine degrees, and therefore firing is seven one
one degrees.
OK, and that defines our the a relative to the.