# Lesson Explainer: Scalars and Vectors Science

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

A physical quantity is something that can be measured. When we make a measurement of a physical quantity, we can express the result of this measurement as a numerical value and a unit.

For example, suppose that we want to measure the temperature of a glass of water. In this case, the physical quantity that we are trying to measure is the quantity “temperature.”

We could measure this temperature using a thermometer. The diagram below shows a thermometer in a glass of water.

The thermometer gives us a value of 40, and we can see from the scale on the thermometer that the measured value has units of degrees Celsius (). Therefore, we can say that we have measured the temperature of the water to be .

Many physical quantities can be fully described using just a value, or magnitude, along with a unit. Temperature is one example of a physical quantity that is fully described this way.

Quantities like this do not have any direction associated with them. Indeed, it is meaningless to try and talk about a direction for temperature.

Consider the following diagram, which shows two thermometers in the same glass of water that we considered earlier.

One of the thermometers (thermometer A) is oriented vertically, as in the previous diagram. The other (thermometer B) is at an angle.

Suppose that we measure the temperature of the water using thermometer A and get a value of . If we then read the temperature from the scale on thermometer B, we will get the same result of . Both thermometers are measuring the same water, and since direction is not a part of the definition of temperature, the angle of the thermometer cannot affect the measured value.

Physical quantities such as temperature, which are fully described using just a magnitude (along with a unit), are known as scalar quantities.

### Definition: Scalar Quantity

A scalar quantity is a quantity that is fully defined by a magnitude.

In the example with the two thermometers, we saw that the direction of the measurement device was irrelevant to the result of the measurement. This is true for any scalar quantity.

It is important to realize that, for scalar quantities, talking about a direction at all is a meaningless thing to do. It is not just that scalar quantities do not need to have a direction in order to be defined; there really is no such thing as a direction for a scalar quantity.

Another scalar quantity is time. Time has a magnitude, which we usually measure in units of seconds, minutes, or hours. However, if someone told us that a particular action took a time of 30 s to perform, it would clearly be a meaningless question to ask them what direction those seconds were in. There is no such thing as a direction of time; it only has a magnitude.

Let us now look at an example problem involving time.

### Example 1: Understanding That Time Is a Scalar Quantity

Two clocks face in different directions. Do the clocks show the same or different measurements of time?

The question shows us two clocks. The left-hand clock is oriented vertically, while the right-hand clock is turned on its side. We are asked whether these two clocks show the same or different measurements of time.

The physical quantity “time” does not have a direction associated with it. It has only a magnitude. In other words, time is a scalar quantity.

Recall that when making a measurement of a physical quantity that has no direction, the angle of a measurement device cannot affect the measurement result.

This means that the angle of a clock cannot affect the measured time.

In this case, both clocks have the hour hand pointing at 3 and the minute hand pointing at 12. This means that both clocks read a time of 3:00. We can notice that this process of reading the time from the clocks was independent of the angle each clock was at.

Therefore, our answer to the question is that the clocks show the same measurement of time.

Let us now consider the physical quantity “distance.” When we talk about the distance traveled by an object, what we are referring to is the length of the path along which the object moves.

Imagine that we have two walkers who both start at the same position and walk as shown in the diagram below.

Now, we can certainly talk about the direction that each of the walkers travels in. We can see from the diagram that walker A travels to the West, while walker B travels to the East.

However, this direction does not enter into our definition of the distance traveled by each walker. Recall that distance is defined as the length of the path. For each of the two walkers, this length is 40 m. Therefore, the distance traveled by each walker is 40 m.

This value completely defines the distance traveled. Since distance is completely defined by a magnitude, along with a unit, then distance must be a scalar quantity.

Besides the total length of the path, the details of the route taken by a moving object do not affect the measurement of the distance traveled. The path taken could be a straight line, or it could be walking around in a circle or any other shape imaginable; the only relevant thing is the length of the path.

For example, we could also consider a third walker, who we will label walker C. The route taken by walker C is shown in the diagram below, along with the routes of walker A and walker B.

We see from the diagram that walker C walks along a route made up of two straight line segments, each with a length of 20 m. The distance traveled by walker C is the length of this path, which is .

Walker C travels in two different directions throughout their route. However, they walk the same distance of 40 m as that traveled by both walker A and walker B. This is possible because the distance traveled is a scalar quantity that does not depend on direction.

Now let us look at the quantity “speed.” Recall that speed is defined as distance traveled per unit of time.

Since we have seen that distance and time are both scalar quantities, we know that neither of these quantities have a direction associated with them. Since speed is defined in terms of two quantities that do not have directions, perhaps it is not surprising that speed does not have a direction either—that is, speed is also a scalar quantity.

Let us consider the same three walkers from earlier. Suppose that each of them walks their 40 m route in a time of 40 s.

All three walkers travel 40 m in 40 s, which means that all the walkers have the same speed of .

The speed of each walker is completely defined by this value and is independent of any details of the route besides the distance. This means that speed has a magnitude but no associated direction and so must be a scalar quantity.

We have seen that both distance and speed are completely defined by a magnitude and a unit; they do not have an associated direction. However, there are some physical quantities that do have a direction. And, in fact, such quantities actually require a direction to be specified in order to be fully defined. We call these quantities, which require both a magnitude and a direction to be fully defined, vector quantities.

### Definition: Vector Quantity

A vector quantity is a quantity that is fully defined by a magnitude and a direction.

Let us have a look at a couple of example questions to help clarify the difference between scalar quantities and vector quantities.

### Example 2: Identifying What Is Common to Scalar Quantities and Vector Quantities

What must a vector quantity have that a scalar quantity must also have?

1. A magnitude
2. A direction

This question is asking us what a vector quantity must have that a scalar quantity also must have. So, let us begin by recalling what things a vector quantity must have.

We can recall that a vector quantity is a quantity that is fully described by a magnitude and a direction. In other words, a vector quantity must have both a magnitude and a direction.

We can notice that “a magnitude” and “a direction” are both possible answers to the question. We need to work out which of these is something that a scalar quantity must have.

We can recall that a scalar quantity is fully described by a magnitude. Therefore, a scalar quantity must have a magnitude but does not have a direction.

Then, our answer to the question is that a vector quantity and a scalar quantity must both have a magnitude. This is the answer given in option A.

### Example 3: Understanding the Difference between Scalar and Vector Quantities

What must a vector quantity have that a scalar quantity cannot have?

1. A magnitude
2. A direction

This question is asking us what a vector quantity must have that a scalar quantity cannot have. We can begin by recalling what things a vector quantity must have.

We can recall that vector quantities are fully defined by a magnitude and a direction. This means that a vector quantity must have both a magnitude and a direction.

We may notice that “a magnitude” and “a direction” are both possible answers given to us in the question. We need to work out which of these answers is something that a scalar quantity cannot have.

We can recall that scalar quantities are fully described by a magnitude. Therefore, a scalar quantity must have a magnitude but cannot have a direction, since it is described fully by the magnitude alone.

Then, our answer to the question is that it is a direction that a vector quantity must have and a scalar quantity cannot have. This is the answer given in option B.

When we talk about the direction of a vector quantity, it is important to be clear what this means.

Consider the two lines shown in the diagram below.

We would say that these two lines have different directions, as they are at different angles to each other.

Now, instead, consider the two lines in the following diagram.

We would say that these two lines have the same direction as each other since they are both oriented horizontally.

Direction can refer to which way a line is oriented, but it also includes which end of the line is considered the “start.” Suppose that instead of lines, we have arrows. This is shown in the diagram below.

These arrows are both oriented horizontally, just as the two horizontal lines we previously considered. However, because the arrows have heads, we would not describe these two arrows as having the same direction. We would say that the red arrow points to the left, while we would say that the blue arrow points to the right. Alternatively, we could say that the direction of the red arrow is to the left and the direction of the blue arrow is to the right.

An arrow shows everything that is meant when we talk about a direction. The angle at which the arrow is drawn tells us the orientation, while the head of the arrow tells us which way along that line the direction is. For this reason, arrows are often used to show vector quantities visually.

We have said that there are some physical quantities that have a direction as well as a magnitude and that these quantities are known as vector quantities. Now let us have a look at a specific example of such a vector quantity: “force.”

Force has a magnitude—the size or the strength of that force. It also has a direction—the direction in which the force is applied. Let us look at an example scenario to see why the direction of a force is an important part of its definition.

Imagine that we have a large box. Two people can each push the box, exerting a force on it. Let us suppose that these two people can push with equal strength—that is, the forces they apply to the box will each have the same magnitude.

Consider the two situations shown in the diagram below.

In the left half of the diagram, the two people are standing on opposite sides of the box and pushing it in opposite directions. In the right half of the diagram, the two people are standing on the same side of the box as each other and both pushing in the same direction.

The forces exerted by each person are shown as arrows on the diagram. Arrows of the same length have been used to show that the forces have the same magnitude. The direction of each arrow tells us the direction of the corresponding force.

The two situations will not result in the same motion of the box. The effect of two people pushing it will clearly be different depending on whether they push in the same direction as each other or in opposite directions. Therefore, the directions of the forces have to be taken into consideration in order to understand the effects of those forces.

This example scenario highlights an important point that applies to all vector quantities. The direction of a vector quantity is not just an extra, optional piece of information. It is an essential part of the definition of the quantity—without knowing the direction, we cannot fully understand the meaning of a vector quantity.

The box in this example also has a mass; mass is a physical quantity that describes how much force is required to accelerate an object. Importantly, the mass of the box is related to how much force is needed to push the box in either horizontal direction. The quantity “mass” has no direction. It only has a magnitude, typically given in units of kilograms. Therefore, mass is a scalar quantity.

It is important not to confuse the physical quantities mass and weight. In everyday speech, we may hear the words “mass” and “weight” used interchangeably. However, in physics, these are two distinct physical quantities.

Unlike mass, weight has a direction. The weight of the box is the force acting on that box as a result of gravity. The direction of this weight force is always downward, toward the earth, regardless of the motion of the box.

The diagram below shows the weight force on the box in three cases. In the left-hand diagram, the box is at rest. In the middle diagram, the box is being pushed to the right. In the right-hand diagram, the box is being pushed to the left. In all three cases, the weight force, which is indicated by a green arrow, acts downward.

The magnitude of the weight gives the strength of this force. Since weight has both a magnitude and a direction, weight is a vector quantity.

We saw earlier that distance is a scalar quantity—it is fully defined by a magnitude. When thinking about distance, we used an example with walkers who each walked different routes starting from the same position. We saw that, despite walking along these different routes, each traveled the same distance because the length of each route was the same.

Let us consider again the walkers that we labeled as walker A and walker B. The routes walked by these walkers are shown in the diagram below.

We saw earlier that each walker travels a distance of 40 m but that each walks in a different direction. Specifically, walker A travels to the West, while walker B travels to the East. This has been indicated in the diagram by arrows pointing from the start position to the end position for each walker.

There is a physical quantity that describes the motion of the walkers in a way that also takes into account the direction in which they walked. This quantity is “displacement.”

The displacement of an object can be represented by an arrow pointing from the object’s start position to the object’s end position, just as we have shown for the two walkers. The length of this arrow shows the magnitude of the displacement. The direction of the arrow shows the direction of the displacement.

Displacement is fully defined by this magnitude and direction, which means that displacement is a vector quantity.

In the case of walker A, we would say that their displacement is 40 m to the West. Meanwhile, for walker B, the displacement is 40 m to the East.

It is important to be clear that the magnitude of the displacement of an object does not necessarily have the same value as the distance traveled by that object.

In the case of walker A and walker B, the motion is in a single straight line. Therefore, the distance traveled by each walker is equal to the length of the arrow from the start position to the end position. In general, for motion along a single straight line, the distance traveled is equal to the magnitude of the displacement.

However, this is not the case for any motion other than that along a single straight line.

Earlier, we also considered a third walker, labeled walker C. The motion of walker C is shown in the diagram below.

The dashed lines on the diagram indicate the route taken by walker C. This route is 20 m East followed by 20 m North. The arrow shows walker C’s displacement. Recall that the magnitude of displacement is defined as the length of an arrow from the start position to the end position. In this case, the length of this arrow is not the same as the length of the route taken. The length of the route taken is 40 m, while the length of the arrow from the start position to the end position is shorter than this.

In general, for any motion that is not along a single straight line, the magnitude of the displacement will be smaller than the distance traveled.

Let us have a look at a couple of example questions that look at the distinction between distance and displacement.

### Example 4: Working Out the Possible Final Positions of an Object That Travels a Given Distance

The car shown is at the center of a circle. The car moves a distance of 30 metres. What could its final position be?

1. Any point within the circle
2. Only points on the circumference of the circle

The question shows us a diagram of a car, initially positioned at the center of a circle of radius 30 m.

We are told that the car moves a distance of 30 m, and we are asked to work out what its final position could be.

Recall that the distance moved by an object is defined as the length of the path taken by that object. Distance is a scalar quantity; this means that it is fully defined by a magnitude, which in this case is the value of the length of the path.

The fact that distance is a scalar quantity means that any other details besides the length of the path are in no way a part of the definition of the distance. In the situation presented in this question, that means that the car could travel any route along any combination of directions, so long as the total length of that path is 30 m.

The furthest final position that the car could reach is if it travels this distance in a single straight line. This is illustrated in the diagram below.

We see that, in this case, the car ends up 30 m away from where it started. The final position of the car is then on the circumference of the circle. There is no possible way for the car to end up outside of this circle, as there is no path with a length of 30 m from the center of the circle to any point outside the circle.

We have shown that a straight line path will mean that if the car travels a distance of 30 m, then it will end up on the circumference of the circle. However, we have also said that any route of length 30 m is possible, so this straight line is not the only option.

Consider some other possibilities, illustrated in the diagram below.

In the left-hand diagram, the car travels in two different directions (15 m left and 15 m up) for a total distance of 30 m. The car ends up at a position inside the circle.

In the middle diagram, the car travels 15 m left followed by 15 m right, giving a total distance of 30 m. It ends up at the same position it started at.

In the right-hand diagram, the car travels along a curved route of total length 30 m, ending up at another point inside the circle.

These examples are just three possibilities out of many. It should be straightforward to convince ourselves that any point inside the circle may be reached by moving a distance of 30 m, by choosing the right path.

Our answer to the question is therefore that the final position of the car may be any point within the circle. This is the answer given in option A.

### Example 5: Understanding the Relationship between Distance and Magnitude of Displacement

The car shown is at the center of a circle. The car moves a distance of 30 metres, and its final position is 30 metres away from its starting position. Did the car move in a single direction?

1. Yes
2. No

The question shows us a diagram with a car at the center of a circle of radius 30 m and asks us to work out whether this car moves in a single direction.

We are told that the car moves a distance of 30 m.

Let us begin by recalling that distance is defined as the length of the path traveled. Distance is a scalar quantity; this means it is fully defined by this length, which is a magnitude.

So, for the car in the question, we know that it traveled along a path with a length of 30 m.

We are also told that the car’s final position is 30 m away from its starting position. What this means is that if we draw a straight arrow pointing from the start position to the end position, the length of that arrow will be 30 m.

Such an arrow is shown in the diagram below.

This arrow represents the displacement of the car. By stating that the car’s final position is 30 m away from its starting position, the question is telling us that the magnitude of the displacement of the car is 30 m. In this case, the magnitude of the displacement has the same value as the distance traveled.

Notice that whichever direction the arrow is drawn, it will extend from the center of the circle to a point on the circumference of the circle. All such points are 30 m away from the center. In other words, because the car ends up 30 m from its starting position, its final position must be somewhere on this circumference.

Consider a particular point on this circumference—we will imagine this to be the car’s final position. In the diagram below, some possible routes from the car’s initial position at the center of the circle to this final position are shown.

Route A shows the car traveling along a single direction. In this case, the route is along the arrow that would represent the car’s displacement and the distance traveled has the same value of 30 m as the magnitude of the displacement.

If either of the other routes, which involve the car moving in multiple different directions, are taken, then the distance traveled must be greater than 30 m. Route A was the straight-line distance between the start and end positions, and any route other than this straight line distance must be longer. The same is true for any other route we could think of.

Therefore, for the distance traveled by the car and the magnitude of the displacement of the car’s final position from its start position to have the same value, the car must have moved in a single direction.

So, our answer to the question of whether the car moved in a single direction is “yes.” This is the answer given in option A.

We have seen that displacement is a vector quantity that is related to the scalar quantity distance. In a similar way, there exists a vector quantity related to the scalar quantity speed. This vector quantity is velocity.

The velocity of an object is the rate of change of that object’s displacement. In other words, velocity is defined as displacement per unit time. Since displacement is a vector quantity and therefore has a direction, it makes sense that velocity will also have a direction.

Let us see how this works by again considering the two walkers, walker A and walker B. We found earlier that if each walker travels a distance of 40 m in a time of 40 s, then the average speed of each walker is 1 m/s.

However, when considering displacement, we also had to take into account the direction in which each walker traveled. Walker A traveled to the West, giving them a final displacement of 40 m to the West from their start position. Meanwhile, walker B traveled to the East, giving them a final displacement of 40 m to the East. The motion of the two walkers is shown in the diagram below.

On the diagram, we have also indicated the velocity of each walker. The velocity is defined by a magnitude equal to their speed, along with a direction equal to the direction of travel. On the diagram, this is shown by the value 1 m/s to give the magnitude of the velocity, along with an arrow indicating the direction.

We would say that the velocity of walker A is 1 m/s to the West, while the velocity of walker B is 1 m/s to the East.

The final physical quantity we shall consider in this explainer is acceleration.

We may previously have encountered acceleration defined as the rate of change of speed of an object. However, strictly speaking, the acceleration of an object should be defined as the rate of change of that object’s velocity.

Since velocity has a direction, then acceleration must also have a direction. Therefore, acceleration must be a vector quantity. The magnitude of an object’s acceleration tells us how rapidly the object’s velocity is changing, while the direction of acceleration tells us which direction this change is in.

Let us consider a specific scenario where we will see why the direction of acceleration is a fundamental part of the definition.

Suppose we have a car moving at a constant velocity along a straight road. Then, suppose we are told that the car accelerates with some magnitude that we will label as .

Two possible outcomes are shown in the diagram below.

Recall that acceleration is a vector quantity, but we were only told the magnitude and not the direction. If the acceleration is in the same direction as the car’s initial velocity, as in the left half of the diagram, it will act to increase the magnitude of the car’s velocity.

However, it could equally be the case that the acceleration is in the opposite direction to the car’s initial velocity. This is shown in the right half of the diagram. In this case, the acceleration acts to reduce the magnitude of the velocity—in other words, the car slows down. When the acceleration acts to reduce the magnitude of the velocity of an object, this is often referred to as a negative acceleration or a deceleration.

What we have seen is that knowing the magnitude of the acceleration is not sufficient information to tell us what will happen to the car. Depending on the direction of that acceleration, the car may either speed up or it may slow down. Therefore, the direction is a fundamental part of the definition of acceleration.

Let us now summarize what we have learned.

### Key Points

• A physical quantity is something that can be measured.
• Physical quantities that are defined completely by a magnitude (along with a unit) are known as scalar quantities.
• Examples of scalar quantities include temperature, time, mass, distance, and speed.
• Physical quantities that are defined completely by a direction and a magnitude (along with a unit) are known as vector quantities.
• Examples of vector quantities include displacement, velocity, force, acceleration, and weight.