Vectors, Scalars, & Displacement: Study with Video Lessons, Practice Problems & Examples

In physics, measurements are categorized as vectors or scalars. Scalars, like mass, time, and temperature, have only magnitude, while vectors, such as force and velocity, include both magnitude and direction. For example, distance is a scalar representing total length traveled, whereas displacement is a vector indicating the shortest path between initial and final positions. The equations for displacement (Δx = x_final - x_initial) and distance (d = total lengths) illustrate these concepts, emphasizing the importance of direction in vector quantities.

Introduction to Vectors and Scalars

Hey, guys. So in this video, I'm going to introduce you to two types of measurements that you'll need to know called vectors and scalars. Let's check it out. So whenever we take measurements in science, we're always going to get the magnitude. For example, let's say you take a thermometer outside and you measure the temperature to 60 degrees Fahrenheit, or you measure a box and you weigh it to be 10 kilograms. The magnitude is really just the size of the measurement. You can think about the size as just the number. And so the size or the number of the measurement is the magnitude. That's 60 and the 10, and then you just have the units. But some measurements are a little bit more specific. Some measurements also give us direction. For example, let's say you're describing where you walked outside and you say, I walked 10 meters to the right. Or let's say you're describing where you were driving, and you say you're going 20 miles an hour to the north. So both of those things are examples of direction. And so let's go through a bunch of examples to make this stuff really, really straightforward. So let's say you weigh an apple and it weighs 5 kilograms. You're measuring the quantity of mass. The 5 kilograms represents the magnitude. But it doesn't make sense to ask which direction those 5 kilograms go. So, we don't have direction there. What about days? Days are 24 hours long, so we're measuring a time there. The 24 hours is our magnitude, but it doesn't make sense to ask which direction those 24 hours go. Now, we already talked about this one. It's 60 degrees outside, so 60 degrees is a measurement of temperature. And we have magnitude, but it doesn't make sense to ask which direction do those 60 degrees go in. So we don't have direction there. Now for this last one here. I pushed with a 100 Newtons to the left. And when you're pushing something, you're measuring the force. The 100 newtons is the magnitude. And here, it actually does make sense to ask which direction you're pushing that could affect things. Are you pushing the box or whatever you're pushing to the right, to the north, to the left? So here we do have a direction. And so in physics, measurements with direction are called vectors. So for example, force has magnitude and direction, so it is a vector. Whereas measurements without direction are called scalars. So for instance, mass, time, and temperature, since they have magnitude only and not the direction, these are examples of scalars. So let's take a look at two sort of related ideas or related measurements. I walked for 10 feet. So let's say you talk to your friend, you say I walked for 10 feet. So that you have the measurement right here, that's the magnitude, but you're not specifying which direction you went in. You could've gone 5 to the right, 5 to the left. You could have gone 10 in whatever direction. And so we don't have the complete idea of that measurement. Whereas, now let's say you talk to your other friend, you say, well, I walk 10 feet toward the east direction. So now you have the magnitude and the direction. So we know that this one's going to be the vector and this one is going to be the scalar. And so in physics, there are two special words for these measurements. I walked for 10 feet is called the distance, the scalar version of it. Whereas the more complete picture, the vector, is called the displacement. And both of these words here answer the question, how far did you go? Now let's take a look at the last two. I drove for 80 miles per hour or I drove for 80 miles per hour to the west. So 80 miles per hour is our magnitude, but it doesn't give us the direction. You could have gone 80 miles per hour to the south, to the west, east, or north. You don't have what the direction is. Versus this sentence over here, I drove 80 miles per hour to the west, gives us more information, a more complete idea of what that motion or measurement is. You have magnitude and direction. So just like the other two, this one's going to be vector and this one's going to be the scalar. And we have two special words for these measurements as well. This one, the scalar is going to be called the speed and the vector is going to be called the velocity. Both of these words here answer the question, how fast did you move or did you go? And, the other thing to remember is that, or one thing to make it easier to remember is that the velocity, v, is the vector and speed with s is the scalar. So v with v, s with s. Hopefully, guys, I've painted a really, really clear picture of the difference between vectors and scalars. So that's it for this one.

Displacement vs. Distance

Hey, guys. In previous videos, we talked about the difference between vectors and scalars and we used displacement and distance as examples of vectors and scalars. We said that there were two similar-sounding words to describe how far something moved and so we're measuring the quantity of length. So in this video, I want to talk about the difference between those and more importantly, I'm going to show you how to calculate each one of these things. So let's check it out. Let's actually just take a look at this example over here. Let's say you had some kind of measuring tape or ruler and you went 10 meters to the right and 6 meters to the left. Well, there are two different numbers that you can get out of these two motions, 10 to the right and 6 to the left. The basic idea is that the distance, which is represented by the letter d, is just the total of all the motions that you do. It's the total of all the lengths traveled, whereas the displacement, on the other hand, is a little bit more specific. It's a change in your position. In physics, your position is just where you are on this number line here. It's represented by the letter x. And one way you can think about the displacement is that it's the shortest path between your initial and final positions. Your initial position is just x0. Your final position is x. These are the two symbols that you'll see. So in this number line here, you started off at x0 and you ended off at x. And the shortest path is just the arrow that connects those two things. So this right here is your displacement. So let's talk about the numbers in this specific example. So you went 10 to the right and 6 to the left, and the total of all the lengths that you travel is just 10 + 6 and that's 16 meters. So notice how this measurement also doesn't have a direction, It's just a magnitude only. So it's a scalar. Whereas the displacement is really just how far you've actually changed from where you started to ending. How far you actually moved. Well, you went 10 to the right and you went 6 backwards, so that means you ended up at 4. Whereas your initial position was 0. So your displacement is just 4 minus 0, which is 4 meters to the right. Another way you can think about this also is that you went 10 and then 6 backwards, so you also get 4. Notice how this measurement has a magnitude and direction, all the distances, and there could be many more than 2. So for example, this was D1, this is 10, this was D2, which is 6, and then you ended up with 16. Whereas your displacement, Δx, which gets a little arrow on top of it because it's a vector, is going to be your final minus your initial position over here. And that's really all there is to it, guys. So let's go ahead and take a look at some examples down here. We're going to be calculating the displacement and the total distance from A to B for each of these situations. So we're going to have x0 equals negative 2 and then x final equals 7. So the shortest path is that arrow that connects them. And so, Δx is just x final minus x initial. So it's just going to be 7 minus negative 2. So be careful with the negative signs. You're going to add those two things together, and we're going to get plus 9 meters to the right. So for the distance, the total distance traveled, that's going to be all the distances involved over here. There could be multiple. And really, there's just one total distance that you traveled, d, is just 9. You went 2 and then 7 this way. Right. So it's 2 to go to 0 and then 7, in the opposite directions. The whole thing is 9. So that means that your total distance is just 9. So notice how these two numbers are the same and that's perfectly fine. They absolutely can be the same number. Let's check out the next example. So now we're actually going to the left in this case. So that's important. We're going from 7 all the way to 3. So our displacement Δx is x final minus x initial. Our x final is 3. Our x initial is 7, and so our displacement is negative 4 meters. So this is also going to be to the left. So the negative sign or to the left also just means the same thing, negative signs. Yeah. So that just means the same thing. And so your distance over here, your total distance is just again, d1, d2, and so on and so forth. Well, there's really only one distance that we traveled from 7 to 3, but the distance is always going to be positive and it's just 4. So that means that our total distance is 4 meters. Notice how this is positive because the direction doesn't matter, it's just the total of all lengths traveled, whereas this has a negative sign because it has a direction. And so finally, let's take a look at our last example. We're going to move from 4 all the way to 10, but then we're actually going to move back to where we started from. So what is the total displacement Δx? It's x final minus x initial. My initial position was actually 4. I went to 10, but then I went all the way backward. So that means that I actually ended up right over here. This is my final position at 4. So my final minus initial is just 4 minus 4, which is just 0. In other words, I haven't displaced anywhere because I went forward and I came back to where I started from. What about the total distance? Well, this is going to be d1 and d2 and so on and so forth. This first distance over here is actually 6 because I'm going from 4 to 10. The second one, when going to 10 to 4, is also another 6. So this is also 6. So let's call that D2. Let's call this D1. And so this is just going to be D total is 6 plus 6 is 12. So even though you literally walked 12 meters, you've actually displaced nothing because you ended up back to where you started from. So this is the distance and that's the displacement. So, really, the displacements can sometimes be negative, as we've seen before, as we've seen here in these examples, but distances are always going to be positive. And in physics, these positive and negative signs are usually just used to indicate the direction. For example, we got negative 4 meters and that just means that we were going to the left. Here, we got positive 9 meters. That just means we were moving to the right. That's it for this one, guys. Let me know if you have any questions.

Starting from a pillar, you run a distance 140m east (the + x-direction), then turn around. 

(a) How far west would you have to walk so that your total distance traveled is 300m?

(b) What is the magnitude and direction of your total displacement?

• A. (a) 160 m, (b) 20 m, west
• B. (a) 160 m, (b) 20 m, east
• C. (a) 580 m, (b) -300 m, west
• D. (a) 440 m, (b) -300 m, east

Here’s what students ask on this topic:

What is the difference between vectors and scalars in physics?

In physics, vectors and scalars are two types of measurements. Scalars have only magnitude, meaning they represent a quantity without any direction. Examples include mass, time, and temperature. Vectors, on the other hand, have both magnitude and direction. Examples of vectors include force, velocity, and displacement. For instance, if you say you walked 10 meters, that's a scalar (distance). But if you say you walked 10 meters to the east, that's a vector (displacement). Scalars are simpler as they only involve numerical values, while vectors provide a more complete description by including direction.

How do you calculate displacement in physics?

Displacement in physics is calculated as the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. The formula for displacement is:

$\mathrm{\Delta x}=x_{\mathrm{final}}-x_{\mathrm{initial}}$

Here, $x_{\mathrm{final}}$ is the final position and $x_{\mathrm{initial}}$ is the initial position. For example, if you move from position 2 meters to 7 meters, your displacement is:

$\mathrm{\Delta x}=7-2=5$ meters.

What is the difference between distance and displacement?

Distance and displacement are both measurements of length, but they differ in their properties. Distance is a scalar quantity that represents the total length traveled, regardless of direction. It is always positive. Displacement, on the other hand, is a vector quantity that represents the shortest path between the initial and final positions, including direction. For example, if you walk 10 meters to the right and then 6 meters to the left, your total distance is 16 meters, but your displacement is 4 meters to the right.

How do you calculate the total distance traveled?

The total distance traveled is the sum of all individual lengths covered during the motion, regardless of direction. It is a scalar quantity and is always positive. For example, if you move 10 meters to the right and then 6 meters to the left, the total distance traveled is:

$d=10+6=16$ meters.

Unlike displacement, distance does not consider the direction of movement, only the total length covered.

What are some examples of scalar and vector quantities?

Scalar quantities have only magnitude and no direction. Examples include:

• Mass (e.g., 5 kg)
• Time (e.g., 24 hours)
• Temperature (e.g., 60 degrees Fahrenheit)

Vector quantities have both magnitude and direction. Examples include:

• Force (e.g., 100 Newtons to the left)
• Velocity (e.g., 80 miles per hour to the west)
• Displacement (e.g., 10 meters to the east)

Understanding the difference between these types of quantities is crucial for solving physics problems accurately.