In this video, we’re talking about scale diagrams. These are diagrams of real objects or quantities that represent those objects according to their accurate proportions. And not only that, but scale diagrams create a connection between some physical quantity and the representation of that quantity in the diagram. For example, in this sketch of the house, one centimeter of distance on the paper might represent one meter of vertical distance on the actual house. That would be one way of setting up a scale between the actual object and our representation of it.
One common type of scale diagram involves using vectors, objects with magnitude as well as direction, as representatives of physical quantities. Often, we use vectors to represent forces, but there are lots of other physical quantities they can depict. For example, displacement is a vector, so is velocity, so is acceleration, and the list goes on.
Now, let’s say we had two vectors like this, and further we know that these vectors are drawn to scale. That means if we were to draw in a grid pattern that the vectors appear on, then the spacing between adjacent grid marks, both in the horizontal direction and in the vertical direction, is consistent. That is, if we were to take a ruler and measure the distance between, say, these two grid marks here, that distance would be the same as the distance between any two adjacent marks anywhere in our diagram, whether we align our ruler vertically or horizontally.
This, by the way, is what wasn’t done in the diagram of the house that we saw in the opening screen. In that case, in the representation of the house, in the diagram of it, the horizontal extent was not the same, it wasn’t equivalent or to scale, with the vertical extent. But in this diagram, with these vectors on this grid, we indeed have a scale diagram.
Now, we said earlier that vectors can represent any number of quantities. These two vectors we’ve drawn here might represent forces in units of newtons, or velocities in units of meters per second, or something else. When we put them on this grid, though, the important thing is not the particular units of the quantities represented by the vectors, but rather it’s the spatial distance between adjacent grid marks on our scale. And often, the way we set this distance is using a ruler.
Consider this ruler that we’ve drawn in. We see that it’s marked out in units of centimeters, and that the distance from one grid mark to the next on our scale is equal to one centimeter on our ruler. For this diagram we’ve drawn then, our scale, whether vertically or horizontally, is that our grid marks are separated by one centimeter. Having a scale diagram lets us combine vectors drawn in it using the diagram. To see how this works, let’s add together, combine, the pink and the orange vectors.
To do this, we’ll first break these vectors up into their horizontal and vertical components. And by drawing in a few axes, we can establish the horizontal direction as well as the vertical. If we focus first on the orange vector, we can draw in the horizontal component of that vector like this. Starting at the origin, we move out along the horizontal axis until we reach the horizontal extent of this vector. And then, on the vertical axis, we do the same thing for its vertical component, starting at the origin and moving up until we reach its vertical extent.
What we’ve drawn in along these axes are the horizontal and vertical components, respectively, of the orange vector. And notice that if we add these two components together, we get the original orange vector. Now, let’s do something similar with the pink vector, breaking it up into its constituent components. The horizontal component of the pink vector starts at the origin and goes out to the right two units on our grid. And then, the vertical component again begins at the origin and moves up four units, or according to our scale four centimeters.
Now, remember that what we want to do is add together graphically the pink and orange vectors. We can do this by adding together their vertical and horizontal components, respectively. To prepare for that, we’ll add a few grid spacings in the vertical and horizontal direction. And then, we can start by taking our two horizontal vectors, the pink and orange components, of those vectors and then arranging them so that they’re end-to-end like this.
Set up this way, we can count the number of grid spacings they cover. Starting at the origin, we count one, two, three, four, five, six, seven grid spacings, which, according to our scale, is seven centimeters. So, when we add the pink and orange vectors together, their horizontal component goes from the origin up to this point here along the horizontal axis.
Now, let’s do a similar addition for the vertical components of these vectors. If we shift these vectors so that they’re now end-to-end along the vertical axis, then once more we can count grid spacings starting at the origin. One, two, three, four, five, six. The combined vertical component of these two vectors, then, will go from the origin up six units on the vertical axis, or six centimeters.
Starting from this mark, we can draw a horizontal line out across our grid. And then, starting on our horizontal axis, seven centimeters to the right of the origin, we can draw a vertical line up from that point. And where these two lines intersect is where our resultant vector will point when it starts from the origin. So, if we add together the pink and orange vectors, the result we get looks like our blue vector. It has a horizontal component of seven centimeters and a vertical component of six centimeters. And we know that because of our grid and because we’ve combined the horizontal and vertical components of the orange and the pink vectors.
Now, not only can we use a scale diagram to add vectors together, but we can also use it to find the direction of vector points. For example, say that we wanted to find the direction of this resultant blue vector. We could define that direction as the angle between the positive horizontal axis here and the blue vector itself. And we could even give that angle a name. Let’s call it 𝜃.
As we think about solving for 𝜃, we can recall that earlier we used a ruler to establish distances on our grid. Those were linear distances, the spacings between grid marks. And now, we want to measure 𝜃, what we could call an angular distance. To do that, we use a tool called a protractor. A protractor is a device typically made of clear plastic that lets us measure angles starting from zero degrees and going up to 180.
When we use a protractor to measure an angle, there are two important things to keep in mind. The first thing is to position the intersection point between the vertical line going to 90 degrees and the horizontal line that goes to zero degrees and 180 so that that intersection point between those two lines goes right on top of the origin of our axes. And the next thing to ensure is that the line on our protractor that goes from 180 degrees to zero degrees is parallel to the horizontal axis of our axes. If we follow these two steps, that will mean our protractor is positioned accurately for measuring our angle 𝜃.
When we shift our protractor into this position, here’s what that would look like. Now, because protractors are typically clear material, we could often see the vectors in the grid lines behind it. But here, we’ve cleared those lines away, just for clarity’s sake. We can see, though, that if we were to continue the line of our blue vector back to the origin, where it truly does begin, that this would be at the intersection point between the 90-degree and zero-degree angle markers on our protractor. And along with this, the protractor is set up so that its horizontal line between zero degrees and 180 is parallel with our horizontal axis.
This means that we can now use the markings on the protractor to start from zero degrees and go up to the blue vector whose angle we want to measure. We would read off this value on the protractor where the blue vector passes right by those markings. And because we’ve started from zero degrees and moved up to that angle, our reading would be equal to the angle we wanted to solve for, 𝜃. We can see that, in essence, we’ve measured the angle between this resultant vector, the blue one, and its horizontal component. We haven’t drawn that component in, but we know it lies along the horizontal axis.
So, on a scale diagram, we can use a ruler to measure distances and a protractor to measure angles. And thanks to the consistency of our scale, that the grid marks are always separated by the same distance, we know that our measurements with a ruler and a protractor will accurately describe our scale diagram. Knowing all this, let’s get a bit of practice now with an example exercise.
Some vectors are drawn to scale on a grid. The green vector is the vertical component of the red vector. The blue vector is the horizontal component of the red vector. What is the angle between the red vector and its horizontal component?
Okay, looking at this diagram, we see this grid with the three vectors drawn in. There’s the red vector, the green one, and the blue one. And we’re told that the green vector is the vertical component of the red, and the blue vector is its horizontal component. This means that if we measure the total vertical extent of the red vector, that length would be indicated by the green vector. And that if we measure the total horizontal extent of the red vector, that length would be indicated by the blue vector.
Our question asks, what is the angle between the red vector and its horizontal component? And we know that another way to say this is, what is the angle between the red vector and the blue vector? If we start with the red vector, this angle would look something like this. We want to solve for what that angle is. And we’ll do it using this tool called a protractor.
This protractor evenly divides out angles between zero degrees on one side and 180 degrees on the other. And we can see that this protractor is positioned so that the place where the blue vector and the green vector cross is right where the 90 degree angle line on the protractor and the zero, or 180-degree angle line intersect. And along with that, the horizontal line connecting zero and 180 degrees on the protractor is lined up parallel with the blue vector. That is, it’s parallel with the horizontal component of the red vector. All this means that our protractor is in the correct position to measure this angle we’re interested in between the red vector and its horizontal component.
As we look at our protractor, though, we may notice that it has two angle scales on it. One scale, the one on the outer edge, goes from zero on the left to 180 degrees on the right. And the other, what we could call the inner scale, goes in the opposite direction, starting at 180 degrees on the left and going to zero degrees on the right. The protractor is set up this way so we can always easily measure from zero degrees up to a positive angle.
The angle that we’ll measure in this case, we can call this angle 𝜃, will also be a positive angle. And we can see it will equal the difference between the angular position of the blue vector and that of the red vector. We could measure 𝜃 starting on the inner measurement scale going from the red line down to zero degrees. But looking at our protractor, we see there’s actually an advantage to using the outermost measurements set.
Notice that on the inner scale on our protractor, the smallest difference between marked out angles on the scale is 10 degrees. But if we look at the outermost scale, we see that the angles are divided up into much smaller increments. For example, let’s consider starting at this particular mark, the one for 50 degrees on our outer measurement scale. Now, if we count the number of tick marks as we go up towards 60 degrees, then we count one tick mark, two, three, four, five. And that marks the halfway point between 50 and 60.
And then, we go up to the sixth tick mark, seventh, eighth, ninth, and then 10th is 60 degrees. So, if there are 10 tick marks evenly spaced between 60 degrees and 50 degrees, that means that the difference between adjacent tick marks, what we could call the resolution of this scale, is one degree. That’s the smallest measurable difference between angles that this scale allows. Well, one degree is much better than 10 degrees. So, let’s use the outermost scale as we measure our angle of interest, 𝜃.
If we look at where our red vector crosses the outermost measurement scale on our protractor, we see that that mark is one, two tick marks above the angle marked out as 130 degrees. And if we look to the left of that, we could call it counterclockwise on our protractor, we see the next marked out angle is 120 degrees. This means the red vector crosses our protractor at two tick marks, or two degrees, below 130 degrees. 130 degrees minus two is 128.
However, that’s not the angle 𝜃 we’re interested in. Rather, 128 degrees would be this angle measurement here, if we measured from zero degrees on our outermost scale up to our red vector. We can see, though, that 𝜃 is the difference between this and 180 degrees. That’s because the largest angle on our outermost measurement scale is indeed 180. Which means that the way to solve for 𝜃 is to take 128 degrees and subtract it from 180 degrees. When we do that, we indeed get this angle marked out in orange, the angle 𝜃. 180 degrees minus 128 degrees is 52 degrees. That is the angle between the red vector and its horizontal component.
Let’s take a moment now to summarize what we’ve learned about scale diagrams. We saw in this lesson that scale diagrams depict vector quantities, like force or displacement or accelerations, according to a consistent marking scale. For example, if this was a vector representing, say, a force, then we saw that we could put this vector on a grid with evenly spaced markings. And that the scale of that grid, the distance between the grid spacings, could be established by a ruler.
We then saw that when vectors are drawn to scale, that means that their magnitudes as well as their directions, in other words, their angles off of, say, a horizontal axis, can be measured on the grid on which they appear. And that measurement happens using a ruler to determine vector magnitude and a protractor to determine vector direction, or angle.