Hi. Welcome back to An Introduction to

Engineering Mechanics. One of the important skills that

engineers have is to take a complex problem or a real world situation and

model it so that we can analyze or perhaps design our system in a different

way and still have our model. were.

We wanted simplest model possible yet we wanted to be giving us accurate or

suitable results and so, last class we talked about modelling a or representing

a force in a 2 dimensions and x and y or a plane just a single plane.

Sometimes, some of the problems that you may have to work with will have to be

used 3 dimensional approach. To be more accurate and to provide you a

better solution. And so, in this case, today, we're going

to look at a 3-d representation in X, Y and Z coordinate system.

So, the learning outcome for today is to express a three-dimensional force in

terms of it's rectangular and components. And then we're also going to make use of

the dot product to find the projection of a force vector.

Vector. So this is the steps that you'll go

through in, in, doing a 3D representation of a force.

You're going to find the position vector along the line of action of force, you're

going to determine a unit vector along that line of action.

And then you're going to mu-, multiply that unit vector by the magnitude of

force to provide you with a force vector. it's easiest to learn this material by

actually going through an example. You can always refer back to this slide

to see the steps that we've taken to, to, to do the problem.

So here I have a, a worksheet, and I'm looking for a 3D force representation of

the force F and then the force P, and Once we have both of those, we can find

the resultant of those 2 forces using the, the parallelogram, parallelogram

rule that we learned in the last module. And so what I'm going to do is, I'm going

to do the forced, F with you together. I'm going to ask that you do force P on

your own and then you also do the resultant on your own since you'll then

know how to do those steps. So the first step is to find the position

vector along the line of action of the force.

In this case, the force F goes from point A up to point B.

And so, we're looking for the position vector from A to B.

And what we do as I, I mentioned here is we walk from tail to head.

The tail of the force is at point a, the head of the line of action for the force

is at point b. And so in going from point a to point b

we so how far do we go in the x direction.

Well in this case we go -3 units in the x direction.

That's -3i. And then we go, + 12 units in the j

direction, or the y direction. So that's +12j.

And then we go three units in the k direction, or +3k.

Okay. And here I've written it again.

Once we have that now we want to find a unit vector in the direction of that

force F. And the way we do that is that we der, we

divide the position vector by its magnitude.

And so the magnitude, the magnitude of the vector AB can be found by taking the

square root of the sum of the squares. So we have the square root of

(-4)^2+(12)^2+(3)^2, and that = 13. And so, our unit vector, which I'll

represent with a E, with a vector designation of the e, so this is eAB is