Video Transcript
In this video, our topic is scalar
and vector quantities. We’re going to learn how to define
these two terms, as well as different ways of combining scalar and vector
quantities. We can see that this sketch here is
hinting at a difference between them. But to clarify things as we get
started, let’s define these two terms.
A scalar quantity, or scalar for
short, is an amount or number of something that’s described completely by a
magnitude or a size. As a simple example of this, if we
had, say, five apples, then the scalar quantity describing the number of apples is
simply five apples. In general, a scalar quantity can
just be a pure number by itself. In this case, that would be
five. Or it can be a number along with a
unit. And in this example, we can think
of the word apples as a unit, the type of thing we’re quantifying.
But let’s think of some other
examples of scalar quantities. Let’s say that we had to go to the
grocery store, and the store was 1.3 kilometers away. That’s a scalar quantity. And in general, this is true of any
distance. Or what about if we made plans to
meet with a friend 30 minutes from now? That’s a time, and all time values
are scalars. The same is true of temperatures
and also speeds. At this point, we may begin to
wonder just what kind of quantities are not scalars. That is, what is left for vector
quantities to describe? But first, a definition.
We can say that vector quantities,
or vectors for short, are quantities that are completely described by a magnitude
and a direction. Thinking back to the example on our
opening screen of those two birds of prey, if they knew that the rabbit was not just
five kilometers away but five kilometers to the east, then they would know the
rabbit’s displacement from themselves. We see that this displacement has a
magnitude, that’s five kilometers, as well as a direction, that’s east, and
therefore is a vector. All displacements share this
quality.
Another type of vector quantity is
all forces. The complete description of a force
involves its magnitude, in this case, seven newtons, as well as the direction the
force acts. So a force is a vector, as is an
acceleration. Whenever we fully define an
acceleration, we describe its magnitude, often in meters per second squared. And we also tell which way it’s
pointing. And as a last example, all
velocities are also vectors. And notice that in this case, we’ve
picked a velocity magnitude that matches our speed. So velocity, which is a vector, is
a speed, a scalar quantity, in some direction. By the way, the vector displacement
and the scalar distance work the same way. The vector displacement consists of
a distance, a scalar, in some direction, in this case, east.
Now that we have the sense for what
scalar and vector quantities are, let’s consider different ways that we might
combine them. To start out, let’s imagine that
we’d like to combine some scalar quantity with another quantity that is also a
scalar. Let’s say that those two quantities
are on the one hand five meters, a distance, and on the other 13 kelvin, a
temperature. We can see right away that even
though both these quantities are scalars, it makes no sense to add them or subtract
one from the other. We could only do this if our two
scalar quantities had the same type of unit. For example, say we wanted to
combine five meters with 27 meters. These are both scalars, and, in
this case, there would be no issue with adding them together or subtracting one from
the other.
We can say that in general, if we’d
like to combine two scalar quantities so long as the units match, we can add or
subtract them. And if we then consider combining
two vector quantities, we’ll find a similar outcome. Say, for example, that we’d like to
combine these two vectors: a force of eight newtons to the west with a velocity of
three meters per second to the right. Just like with our two scalar
quantities of this similar type, we’re not able to add these vectors together or
subtract one from the other because they’re also of different types. The quickest way to see that is to
notice that they have different units, newtons and meters per second.
So, even though both these
quantities are vectors, we can’t add and subtract them because of the units
mismatch. But if they did have the same sort
of units, say this force here and this one we’ve just defined, then combining them
through either addition or subtraction would make sense. So then, if we want to combine a
scalar with a scalar or a vector with a vector, we can add or subtract these pairs
of quantities so long as their units agree. And now let’s consider a third
possibility for combining scalars and vectors.
What if we wanted to combine a
scalar quantity with a vector quantity? For example, say we wanted to
combine a time, say of 16 seconds, with an acceleration vector, say of two meters
per second squared to the left. With these examples, it’s clear to
us right away that we can’t simply add or subtract these scalar and vector
quantities. But to explore this idea a bit,
let’s say that the scalar quantity and vector quantity we picked actually had the
same type of units. For example, what if we wanted to
combine a speed, which is a scalar with units of meters per second, with velocity,
which is a vector but yet has the same units? It turns out that even in this case
of unit agreement, we can’t add or subtract a scalar and a vector.
Imagine, for example, trying to add
three meters per second to four meters per second to the north. Because our vector quantity has a
direction, we really wouldn’t know how to combine that direction aspect of it with a
nondirectional scalar. We just don’t have that information
about the speed, and so we can’t add it to a velocity. Perhaps surprisingly, this doesn’t
mean that there’s no way to combine a scalar quantity with a vector quantity. That’s because the mathematical
operations of multiplication and division are still open to us.
And as strange as it may seem,
these operations between a scalar and a vector, in general, are allowed. As an example, say that we divide
our acceleration, a vector, by this amount of time, a scalar. The outcome of this would be a
fraction, two sixteenths, or equivalently one-eighth, of a meter per second squared
per second acting to the left. Now, a meter per second squared per
second could also be written as a meter per second cubed. And so what we have here is
essentially a time rate of change of an acceleration, that is, how much some
acceleration in meters per second squared changes every second.
Now, this is definitely a strange
unit and not one we usually encounter, but that doesn’t mean it’s disallowed or
impossible. We could imagine a scenario where
an acceleration changes over some amount of time. And so, even though perhaps very
uncommon, this way of combining these two quantities is permitted. Further examples could show us that
just like division is allowed, so is multiplication between a scalar and vector
quantity. So we’ll summarize this this
way. When we want to combine a scalar
with a vector, we can’t add or subtract them, but we can multiply or divide.
Now, one last point we can make
about combining scalars and vectors is that when we’re considering bringing a vector
and a vector together, combining them either through addition or subtraction, then
if that is allowed — that is, if the units match — then we can do this graphically
as well as arithmetically. To see how this works, let’s clear
a bit of space on screen and then draw a pair of coordinate axes. Now, what we want to do here is
graphically combine two vectors, which we’ll say are of forces. We’ve already got one force eight
newtons to the west, and so let’s define another. Say that it’s two newtons to the
south.
These units of newtons and the
directions attached to them, west and south, show us how we can label the axes on
our graph. Let’s say that this direction is to
the north, this direction is to the east and that our axes are marked out in units
of newtons like this. Now, when we go to plot our vector
of eight newtons to the west on this graph, we see right away that we’ll need to
extend our axis to the west. But once we’ve done that, we can
then plot out our eight-newton westward-pointing force. It starts at the origin and then
goes eight units, eight newtons, in a western direction. Next, we can sketch in our
two-newton to-the-south force. That will look like this on our
graph.
Now, we said earlier that if we
have two vectors and their units agree, then we can add or subtract them. Since these vectors are both
forces, they meet that condition of having the same units. So we should be able to, for
example, add them together. And when vectors are plotted
together on the same graph, like these are, we use what’s called the tip-to-tail
method to do this. That method works like this. Starting with either one of our two
vectors, we locate the tip of that vector. Let’s say we start with our
eight-newton westward vector, which means the tip of that is right here. So, that is our tip in the
tip-to-tail method.
Then, what we do is locate the tail
of the other vector. That’s that point right there. And we translate or shift the
second vector so that its tail lies on top of the tip of the first one. So that would involve moving our
gold vector, the one that points two newtons to the south, like this. Now that our vectors are arranged
tip to tail, we start at the beginning, the tail of the very first one, that’s at
the origin. And we draw a vector from that
point to the tip of our second vector. So we can say that if our first
vector was vector 𝐀 and the second one was vector 𝐁, then this green vector we’ve
just drawn in is equal to 𝐀 plus 𝐁. So then, when we’re combining
vectors, it’s possible not just to do it algebraically, but also graphically.
Knowing all this about scalar and
vector quantities, let’s get a bit of practice of these ideas through an
example.
Which of the following is a vector
quantity? (A) Energy, (B) pressure, (C)
potential difference, (D) force, (E) charge.
Okay, so the idea here is that one
of these five quantities is a vector. And we can recall that a vector
quantity is defined as having both magnitude as well as direction. And specifically, it’s the fact
that a vector has a direction associated with it that sets it apart from what’s
called a scalar quantity, which is a quantity that only has magnitude. So, as we look over our different
answer options, let’s consider examples of each one of these quantities and see if
we find any directions along with magnitudes.
Starting out with choice (A),
energy, an example amount of energy might be this: 12 joules. When it comes to pressures, the
standard unit for reporting a pressure in is the pascal. The standard unit for reporting a
potential difference is the volt. We might have, say, 32 volts. While the SI based unit of force is
the newton, we might have a force say of 13 newtons to the right. And amounts of charge are typically
reported in units of coulombs. We could have 0.3 coulombs of
charge.
In all of these example quantities,
we see that there are magnitudes: 12 joules, 3.1 pascals, 32 volts, and so on. But by our definition of what a
vector quantity is, magnitude alone is not enough to create a vector. We also need a direction associated
with that magnitude. Requiring this to be true, we see
that only one of our five options has a direction. That’s the force, which is a force
of 13 newtons to the right. Here then we have a magnitude, 13
newtons, and a direction. And none of the other quantities
have a direction. So we’ll choose answer option (D),
force, as the only vector quantity listed here.
Let’s look now at a second example
exercise.
If an area is multiplied by a
length, is the resultant quantity a vector quantity or a scalar quantity?
All right, so we’re talking about
multiplying an area by a length. And we could imagine, say, that
this is our area and that this here is our length. When we multiply this area by this
length, we’re going to get some result, and we want to know whether that’s a vector
quantity or a scalar quantity. Here’s the difference between the
two. A scalar quantity, or scalar for
short, has a magnitude only, while a vector quantity, or a vector, has both
magnitude and direction. So, to figure out whether our area
multiplied by a length is a scalar or a vector, we’ll need to see whether it has a
magnitude only or a magnitude along with a direction.
The product of our area and our
length is this volume 𝑉. And so now we ask ourselves, does
this volume have a magnitude only or does it have a magnitude along with a
direction? Put that way, we can see that this
volume does have a magnitude, that is, a size, that’s equal to whatever 𝐴 times 𝐿
is. But there’s no particular direction
in which this volume or any volume points. So here, we do have a magnitude,
but we don’t have a direction. Based on our definitions of scalar
and vector quantities, that then answers our question. If an area is multiplied by a
length, the resultant quantity is a scalar quantity.
Let’s summarize now what we’ve
learned about scalar and vector quantities. In this lesson, we defined scalar
and vector quantities. And we saw that the vital
difference between them is that a scalar quantity has a magnitude only, while a
vector quantity has both magnitude and direction. We saw further that a scalar and a
scalar or a vector and a vector can be added or subtracted if their units agree. Related to this, we saw that a
scalar quantity can multiply or divide a vector. And lastly, we saw that two or more
vectors can be combined graphically using what’s called the tip-to-tail method. This is a summary of scalar and
vector quantities.
