Sometimes, we are given forces that’s from one point to another,
so for example, a force along a rope. We need a way to express that force in cartesian form.
To do that, we just need to follow a set of steps and you’ll always get your answer.
To find the force in cartesian form, we need to know a few things. First,
a position vector. A position vector is simply a vector directed from one point to another.
We represent position vectors using the letter r. We can find the magnitude of this position
vector. We do that by taking each component, squaring it, and then taking the square root.
Now if we divide each component of our position vector by the magnitude,
the result is a unit vector. So let’s move onto some examples to see how we can actually do this.
Let’s take a look at this problem where we need to find the resultant force and the coordinate
direction angles. So we see that this is a force directed along a line, since both forces start
at point A and one goes to point B and another to point C. To find the answers, we need to express
each force in cartesian form. So to do that, the first step is to write down the locations of all
the points of interest. We will start with point A. Point A is at the origin, so x and y components
are zero, but it’s 6 meters up, which means the zed component will be 6. Next, point B. It’s 2m
in the x direction, 3 meters in the negative y direction, and since it’s at the ground level,
the zed component is 0. Lastly, we have point C. That’s 3 m in the x direction, 2 meters in
the y direction and 0 m in the zed direction. The second step is to write position vectors.
So it’s just a vector that starts where the force starts, and then ends where the force
ends. We will first look at force FB. Notice how force FB starts at A and ends at point
B. So to find the position vector, we just subtract the position of A from the position of B.
So we are subtracting each component of A from B. Just a tip, when we look at the way we wrote rab,
B is the last letter. So when we subtract, we actually write B first, and A comes second,
so we are subtracting A from B. If you keep that in mind, you won’t mix them up.
Let’s simplify. Let’s do the same for force FC. So it starts at A and ends at C,
which means we subtract the position of A from C. Now we will find the magnitude of each of these
position vectors. We do that by taking each term, squaring it, adding it all together
and then taking the square root of it. Next, rac. By the way, these magnitudes we found
is the distance from points A to B and A to C. In other words, the length of each rope.
Now that we have the magnitude, we need to find the unit vector.
We do that by taking each term in our position vector and dividing it by the magnitude.
So the last step in expressing our force in cartesian form is to multiply it by
the unit vector. So for force FB, we are just multiplying each term in the unit
vector by 560 N. Let’s simplify. Next, we have force FC. So again,
all we do is multiplying the magnitude of the force by the unit vector.
So now that we have our forces expressed in cartesian form, we can add them together to
get the resultant force. We just add I components to I components, j components to j components and
k components to k components. The magnitude of this force
can be found the same way we did before. To find the coordinate direction angles,
we can use these equations. If we take the inverse of cosine, we get our angles.
Those are our answers. If these equations are unfamiliar to you, please check the description
for the previous video about the addition of cartesian vectors where I discuss them in detail.
Let’s take a look at this question where we need to express each force in the ropes
in cartesian form. So the first step is to write down the locations of points A, B, C,
D, and E. First point A. Next, point B. Now point C. Point D is next. And lastly, we got point E.
Now we have 3 cables, so we need 3 position vectors.
The first will be for force FB, which goes from A to B.
Next, we got force FC, which goes from A to C. Lastly, we have force FE, which goes from D to E.
The next step is to calculate the magnitude of each of the position vectors.
Now that we have the magnitude, we can find the unit vectors. So remember, all we do is divide
each term in the position vector by the magnitude. The last step is to multiply the force by the
unit vector. First force FB. Let’s simplify. Next, force FC. And lastly, we have force FE.
Those are our answers. Let’s take a look at one last example where
we have a force already given to us in cartesian form and we need to find the location of point B.
So first, let’s write down the locations of points A and B. So when we don’t know the value,
we will just use the x, y and zed letters to represent the locations.
Now we will write a position vector from A to B. Let’s simplify.
Notice that we are given the length of the rope in the question, or in other words, the
magnitude. So the unit vector would be each term in the position vector divided by the magnitude.
Now let’s look at the force given to us in Cartesian form.
We can find the magnitude of this force using our usual method.
So now, we know that a force in cartesian form is simply force times the unit vector,
and we are already given the cartesian form of the force. So let’s write it down.
To solve, all we need to do is equate the components. So I components to I components,
j components to j components and k components to k components. Let’s solve and those are our answers.
That should cover the types of problems you will face when it comes to forces directed
along a line. I hope this helped, and if it did, please consider sharing it with your friends and
classmates, they too might find it helpful. Thanks for watching and best of luck with your studies.