Let’s talk about vectors, and how we can add
force vectors to figure out the resultant force. You probably know that a vector is simply a
physical quantity that has both a magnitude and a direction. We show vectors using arrows like this.
So in this example, we are showing the velocity of a car. The length of the arrow represents the
magnitude of the vector, so the longer the arrow, the bigger the magnitude and vice versa. The angle
is what gives the direction. So in this example, we see that the arrow is 30 degrees away from the
x-axis. In other words, it gives us the direction of it’s line of action. Now let’s move onto
forces. Forces are also shown using arrows. For example, here, we see 2 forces applied
to this object. When we have multiple forces applied to an object, we can figure out
something called the resultant force. All it means is that it’s the addition of all the
vectors represented by a single arrow. So really, we are taking multiple arrows and showing all
of them as 1 arrow. The resultant force shows us the same effect on the object that the original
forces do. We can do this in a few ways but in this video, we’re going to use something called
the parallelogram law of addition. In the next few examples, we will go over how to use it. But
before we do that, to use the parallelogram law of addition, you should know how to use the sine
law and the cosine law, since pretty much all of it is just solving triangles. Without further
ado, let’s get started with some examples. Let’s take a look at this problem. There are
2 forces applied to the object, and we need to find the magnitude of the resultant force and
its direction. We will use the parallelogram law of addition to solve this problem. The first
step is to draw a line that is parallel to the 700 N force. Next, a line that’s parallel
to the 450 N force. Notice how this creates a parallelogram. You can see where the name for
this law comes from. Now if we draw another line, starting from the origin to where these 2 lines
intersect, that’s our resultant force vector. It is the diagonal of the parallelogram. Notice
also that we have 2 triangles formed. You can solve this problem using either of the triangles.
I am going to pick the top one and move it out. So one side of the triangle is 450 N and the other
side is 700 N. To find the remaining side, we can use the law of cosines. But to use it, we actually
need an angle inside the triangle. The easiest one to figure out is actually the top angle. If we
draw a line parallel to the x-axis at the top, then that whole angle has to be 60 degrees,
because it’s an alternate interior angle. Now we know that the 700 N force is 15 degrees below the
x-axis, which means the angle between the x-axis parallel line and the 700 N parallel line is 15
degrees at the top as well. So if we subtract one from the other, we get 45 degrees. Now we can
apply the law of cosines. Let’s solve. Just this value isn’t a complete answer, we need an angle
to show the direction of the resultant force since it’s a vector. So what we need is to find
the angle it makes with respect to the x-axis. So first, let’s find the bottom angle in our
triangle. For that, we can again use the law of cosines. Solving gives us the angle inside the
triangle. To get the full angle from the x-axis, all we need to do is add the 60 degrees. So the
answer to our question is both of these parts. Let’s take a look at this question where we need
to figure out the angle between the 2 forces so that the resultant force would have a magnitude of
800 N. So the first step is to draw the parallel lines, starting with the 400 N force. Now for
the 600 N force. Next we draw our resultant force to the point of intersection. The goal of
this problem is to figure out this theta value. But notice how this theta value plus the top
angle must equal 180 degrees since these are consecutive interior angles. In other words, the
angle at the top is 180 degrees minus theta. Now let’s bring our triangle out. We can figure out
the theta value by writing the law of cosines. Let’s simplify. If we take the inverse
of cosine, then we can solve for theta. So for the resultant force to have a magnitude
of 800 N, the angle has to be 75 degrees. Let’s take a look at one last example.
In this question, we need to figure out the magnitude of the resultant force and the
angle theta if the resultant force is directed vertically upwards. In other words, we need
the resultant force to point up on the y-axis. So the first step is to draw our parallel
lines, starting with our 500 N force. Next, the 600 N force. Now we can draw our
resultant force, which is straight upwards. We can easily figure out the top angle since the
30 degree angle is an alternate interior angle. Now we can bring our triangle out. For this
example, let’s use the law of sines. Let’s write it down. Now we can solve for theta.
Let’s figure out the last angle left inside the triangle.
Using this, we can write another law of sines to figure
out the resultant force and that’s our answer. That should cover the types of problems you will
face when it comes to finding resultant forces and the addition of forces. In the next video,
we will cover how to break forces in the x and y plane into x and y components. Thanks for
watching and best of luck with your studies!