Lesson Video: Scalars and Vectors Science

In this video, we will learn how to define physical quantities as scalars or vectors depending on whether they have a direction.

11:04

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.

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