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