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
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
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.