Video Transcript
In this video, we will learn how to
define physical quantities as scalars or vectors, depending on whether they have a
direction. Let’s begin by thinking about what
a physical quantity is. A physical quantity is something
that can be measured. Examples of physical quantities
include mass, length, force, and so on. These are things we can measure,
and therefore they’re physical quantities. There are two types or categories
of physical quantity. Scalars are physical quantities
that possess only magnitude. On the other hand, vectors are
physical quantities that possess magnitude and direction.
Every physical quantity, mass,
length, and so on, can be categorized as either a scalar or a vector. For example, let’s consider
length. Say that we have some object and we
want to measure its length. Imagine that we do make this
measurement and we get a result of 2.4 meters. This length clearly has a
magnitude, or a size. One thing it doesn’t have, though,
is a direction. This measured length has a
magnitude, or size, but no direction to it. We can say then that length is a
scalar physical quantity.
Now let’s consider the physical
quantity of force. Say that we have a box at rest on
the ground and that we exert a force of 60 newtons on the box like this. This force has a magnitude. It’s 60 newtons. But then it also has a direction,
as indicated by this arrow. The force is pointing to the
right. So this force and indeed all forces
have a magnitude as well as a direction. That tells us that force is a
vector physical quantity.
Let’s now consider the rest of
these physical quantities in this list and see how to categorize them whether as
scalars or vectors. If we measure the mass of some
object, we’ll get a single number, in this case 7.4 kilograms. This measurement has a magnitude
but no direction. Mass is an example of a scalar
physical quantity. We’ve already looked at length and
force, so now let’s consider speed. Say that a runner is measured to
have a speed of three meters per second. This measurement definitely has a
magnitude, a size, but the measurement itself has no associated direction. We’ve drawn the runner facing to
the right. That implies a direction. But if we only look at the measured
quantity itself, the speed, that by itself doesn’t tell us which way the runner is
headed. Speed, then, is a scalar physical
quantity.
But now let’s think about a closely
related quantity, velocity. A measurement of the runner’s
velocity would include the runner’s direction. We could say a velocity measurement
includes a speed as well as a direction. In general, velocity is a vector
physical quantity. We’ve seen that speed and velocity
are closely related physical quantities. It turns out there’s another pair
of quantities like this, distance and displacement. To see how they relate, imagine
that we have a walker and that this walker covers a distance of one kilometer. Distance is a physical quantity,
something we can measure, and it has a magnitude, or size, but no direction. So distance is a scalar physical
quantity or just scalar for short.
But say that we then add some
information. Say we know that the walker covered
one kilometer while walking toward the east. This is a measure of the walker’s
displacement, and we can see that this includes a magnitude as well as a
direction. Distance and displacement are
similar, and they’re often confused for one another. But here we see that they are
different. Distance is a scalar, while
displacement is a vector. Remember that the same thing
happened with speed and velocity. These quantities are related and
sometimes thought of as being the same, while really they’re not. Speed is a scalar quantity, while a
velocity includes a speed but also a direction. It’s a vector. Knowing now about these two pairs
of terms, let’s look at a few scalar and vector examples.
Which of the following best states
what a physical quantity is? (A) A physical quantity is a
measurement of something. (B) A physical quantity is
something that can be measured.
These two answer options may seem
to be saying the same thing. They are different, though. Say that we have some physical
object like this. This object has various properties
like mass or speed or height. And say that we were to measure
these particular quantities. Our results might be as
follows. And now notice that we have two
columns. The first column shows the thing
that is being measured, and the second column shows the measurement of that
thing. Answer option (A) is talking about
the measurement itself. That would be our second
column. Answer choice (A) is saying, for
example, that 8.1 kilograms is itself a physical quantity, so as zero meters per
second, and so on.
Option (B), on the other hand, is
talking about the other column, mass and speed and height in general. This is the best definition of a
physical quantity. Rather than being the measurement
of something itself, a physical quantity is something that can be measured.
Let’s look now at another
example.
What must a vector quantity have
that a scalar quantity cannot have? (A) A direction, (B) a
magnitude.
All physical quantities like
acceleration or distance or force can be divided into two categories. Both scalar and vector quantities
have a magnitude, but vectors have something scalars do not: a direction. This is how we define the
difference between scalars and vectors. A direction is something a vector
must have and a scalar cannot have. We choose answer option (A).
Let’s now look at an example
describing the difference between distance and displacement.
Fill in the blanks: A displacement
is a distance in a particular direction. Displacement is the blank quantity
of change of blank. (A) Scalar, speed. (B) Vector, speed. (C) Scalar, position. Or (D) vector, position.
We want to figure out which of
these answer choices correctly fills in the two blanks in our sentence. That sentence is about
displacement, and we’re told that a displacement is a distance in a particular
direction. So if we were given a distance, say
a distance of five kilometers, and also told a direction to go along with the
distance, then taking these two things together, we have a displacement, a distance
in a particular direction. The first blank in our sentence
will be filled either with the word vector or scalar. A vector is a physical quantity
that has magnitude as well as direction. A scalar, however, has magnitude
but no direction. Since displacement has magnitude
and direction, it’s a vector quantity. We can fill in the first blank in
our sentence and cross off answer options (A) and (C).
To fill in the second blank, we’ll
need to know whether displacement is a change of speed or a change of position. Working with our example
displacement of five kilometers to the north, an object could be displaced this
amount by starting out, say, here and then moving a distance of five kilometers to
the north. So at first, the object’s position
was here, and then after being displaced, its position was here. Displacement then has to do with
change of position rather than change of speed. We choose answer (D). And our completed sentence reads
“Displacement is the vector quantity of change of position.”
Let’s look now at one last
example.
Fill in the blanks: It is possible
to move along the length of the blank arrow without blank direction. This is blank possible moving along
the length of the blank arrow.
Each one of our answer options
lists four words or phrases that can fill in these four blanks. The sentences with these blanks
refer to the red and green arrows we see. In particular, we’re thinking about
direction as we move along the length of these arrows. Say that we begin at the tail of
the red arrow here. If we move along the arrow, at this
instant, we’d be moving to the right. At a later instant though, farther
along the arrow’s length, we’d be moving in a slightly different direction. In fact, as we move along the
length of the red arrow, our direction is constantly changing.
On the other hand, if we start at
the tail of the green vector, when we move along its length, we find we’re always
headed in the same direction. The green arrow is straight, so our
direction is constant. Looking at the blanks in our first
sentence, we see the first blank must be filled in either by the word red or
green. The second blank will be filled in
either by this phrase, “having a,” or by the word “changing.” These options refer to the
direction as we move along the specified arrow.
For option (A) to make sense as an
answer choice, it would have to be true that we could move along at least one of
these arrows without having a direction. We’ve seen, though, that that’s not
the case. We always have a direction, even if
that direction changes as it did when we moved along the red arrow. This tells us that the word that
completes the second blank in our first sentence must be the word changing. So this sentence now reads “It is
possible to move along the length of the blank arrow without changing
direction.” We’ve seen that this is true for
the green arrow, but not for the red arrow. Therefore, the first blank in this
sentence we fill in with the word green. And we can cross out answer choice
(D).
Looking at our remaining two answer
options, we see that the word “red” must go in the last blank in our second
sentence. This second sentence then reads
“This is blank possible moving along the length of the red arrow.” The question then is, is it
possible to move along the length of the red arrow without changing direction like
it is with the green arrow? We’ve seen that it’s not, that as
we move along the length of the red arrow, our direction changes constantly. Therefore, we say “This is not
possible moving along the length of the red arrow.” The correct answer is option
(B). And our completed sentences read
“It is possible to move along the length of the green arrow without changing
direction. This is not possible moving along
the length of the red arrow.”
Let’s now finish our lesson by
reviewing a few key points. In this video, we saw that a
physical quantity is something that can be measured like mass or speed or force. A physical quantity is either a
scalar or a vector quantity. A scalar quantity has magnitude and
no direction, examples of this include speed or distance, while a vector quantity
has both magnitude and direction. Examples of vectors include
displacement and velocity. This is a summary of scalars and
vectors.
