# 8.3: What you don’t know about vectors may surprise you!

- Page ID
- 3398

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Remember that a scalar has only a magnitude while a vector has both a magnitude and a direction. The following video (12:33) makes this difference clear.

**Scalars and Vectors**

**Click here for transcript of the Scalars and Vectors video.**-
Hi it's Mr. Andersen and right now I'm actually playing Angry Birds. Angry Birds is a video game where you get to launch Angry Birds at these pig type characters. I like it for two reasons. Number one it's addictive, but number two it deals with physics and a lot of my favorite games deal with physics. So let's go to level two. And so, what I'm going to talk about today are vectors and scalars, and vectors and scalars are ways that we measure quantities in physics. Angry Birds would be a really boring game if I just use scalars because if I just use scalars I would input the speed of the bird and then I would just let it go, and it'd be boring because I wouldn't be able to vary the direction. And so Angry Birds I can vary the direction and let me try to skip this off of... nice. I can try to skip it off and and kill enough of these pigs at once. Now I could play this for the whole 10 minutes but that would probably be a waste of time. So, what I want to do is talk about scalars and vector quantities. Scalar and vector quantities I wanted to start with them at the beginning of physics because sometimes we get two vectors and people get confused and don't understand where did they come from. So, we have quantities that we measure in science especially in physics and we give numbers and units to those, but they come in two different types and those are scalar and vector. To kind of talk about the difference between the two, a scalar quantity is going to be a quantity where we just measure the magnitude, and so an example of a scalar quantity could be speed. So when you measure the speed of something and I say how fast does your car go, you might say that my car goes 109 miles per hour. Or, if you're a physics teacher you might say that my bike goes, I don't know like nine point six meters per second, and so this is going to be speed and the reason it's a scalar quantity is it simply gives me a magnitude. How fast, how far, how big, how quick. All those things are scalar quantities. What's missing from a scalar quantity is direction, and so vector quantities are going to tell you the not only the magnitude, but they're also going to tell you what direction that magnitude is in. So, let me use a different color maybe. Example of a vector quantity would be velocity, and so in science it's really important that we make this distinction between speed and velocity. Speed is just how fast something is going, but velocity is also going to contain the direction. In other words I could say that my bike is going 9.08 m/s West. Or, I could say this pen is being thrown with initial velocity of two point eight meters per second up or in the positive. And so, once we add direction to a quantity now we have a vector. Now you might think to yourself that's kind of nitpicky. Why do we care what direction that were flowing in and I have a demonstration that will kind of show you the importance of that, but a good example would be acceleration. So what is acceleration? Acceleration is simply change in velocity over time and so acceleration is going to be the change in velocity over time. and so I could ask you a question like this. let's say a car is driving down a road and it's going 23 meters per second and it stays at 23 meters per second. Is it accelerating? And you would say no of course it's not. Let's say it goes around a corner and during that movement around the corner it stays at 23 miles per hour. Well what would happen to the scalar quantity of speed around the corner? It would still be 23 meters per second, and so if you're using scalar quantities we'd have to say that it's not accelerating, but since the velocity is a vector if you're going 23 miles an hour and you go around a corner are you accelerating. Yeah, because you're not changing the magnitude of your speed, but you're clearly changing the direction and so a change in velocity is going to be acceleration. And so you are accelerating when you go around a corner. And so that be an example of why in physics, I'm not trying to be nitpicky I'm just saying that you have to understand the difference between a scalar quantity and then which is just magnitude, and a vector which is magnitude and direction. There's a review at the end of this minute video, and so I'll have you go through a bunch of these and so we'll identify a number of them, but for now I want to give you a little demonstration. To show you the importance of a scalar and vector quantities. So what I have here is a one thousand gram weight or one kilogram weight. It's suspended from a scale and I don't know if you can read that on there but the scale measures the number of grams. And so, if this is a thousand grams and this measures the number of grams and it's scaled right it should say and it does about a thousand grams is, is the weight of this. Now a question I could ask you is this, let's say I bring another scale and so I'm going to attach another scale to it. And so if we had one mass that had a mass of a thousand grams, and now I have two scales that are bearing the weight of that and I lift them directly up, what should what should each of the scales read. And if you're thinking well it's a thousand gram so each one should read 500 grams let me try it. The right answer is, yeah. Each of the scales ray right at about five hundred grands and so that should make sense to you. In other words 500 + 500 is a thousand so we have the force down of the weight force of tension that's holding these in position, and so we should be good to go. The problem becomes when I start to change the angle and so what I'm going to do and I'm sure this will go off screen, is I'm going to start to to hold these at a different angle. and so what if they look right here and now find that it's a six hundred and so this one is at 600 as well. and so as I increase the angle like this will find that that will increase as well and so when I get it to an angle like this I have a thousand gram weight and it's being supported by two scales now that are reading a thousand. and it's going to vary as I come back to here and if you do any weight lifting you understand kind of how that works. So the question becomes how do we do math? The problem with this then is the the numbers don't add up. And so, if I've got a 500 gram way excuse me a thousand gram weight being supported by two scales it made sense that it was weighing five hundred each. But now we all the sudden have a thousand gram weight being supported by two scales that are reading thousand and so this doesn't make sense or the math doesn't make sense. And the reason why is that you're trying to solve the problem from a scalar perspective, and you'll never be able to get the right answer because it's going to change its going to change depending on the angle that we lift them at. So, to understand this in a a vector method, and we'll get way into detail so I just kind of wanted to touch on it for just a second. What we had was a weight so we'll say there's a weight like this and will say that's a thousand gram weight and then we have two scales and each of those scales are pulling at 500 grams. So, if you add the vectors up, so this is 1 vector and this is another vector, so each of these are 500 grams so I make them 500 in length. Then we balance out in other words you have the balancing of this weight with these two weights that are on top of it. Now if we go to the vector problem the vector problem again we had a thousand gram weights a thousand grams in the middle, and then we had a force in this direction of a thousand and a force in that direction of the thousand. So we had the force down of a thousand, but we had a force of a thousand in this direction and a force of a thousand in that direction. And so, if you start to look of it at it like a vector quantity imagine this that we've gotta weight right here but you have to have two people pulling on it and so it's like this tug-of-war where it's not just in one direction but it's actually in two. And so you can start to see how these forces are going to balance out, but only if we look at it from the vector perspective. Let me show you what that would actually look like. So if we put these tails up this would be that force down of a thousand grams. This would be the force of the weight, but we also had a force in this direction so I'm doing the same rule where I'm lining up my vector from the tail to the tip and the tail to the tip. And so that diagram that I had in the last slide I'm actually moving this one force and you can see that they all sum up to 0. and so the reason I like to start talking about vectors and scalars with this problem is that you can never solve the problem if you're going to go at it from a scalar perspective. and we're going to do some really cool problems let's say I'm sliding a box across the floor, but how often do you slide a box across the floor and actually pull it straight across like that? if you're like me you're pulling a sled or something you normally pulling it at an angle and once we start playing at an angle becomes a totally different for us and we can't solve problems in the scalar way we have to go and solve it from the vector perspective and so that's the importance of vectors. on now it's a huge thing. So there are lots of things that we can measure in physics and so what I'm going to try to do hopefully can get this right is go through and circle all the scalar quantities and then go back and circle all the vector quantities. And so if you're watching this video a good thing to do would be pause right now and then you go through in and circle the ones that you think are scalar and vector, and then we'll see if we match up the end. Scalar quantities remember is simply going to be magnitude. And so the question I always ask myself when I'm doing this is, ok does it have a direction? Length is simply the length of a side of something, so I would put that in the scalar perspective. This is kind of philosophical, does time have a direction? I would say no. Acceleration we already talked about that. That's changing in velocity. What about density, the density of something. That definitely is a scalar quantity. If I say the density of that is 12.8 grams per cubic centimeter North that doesn't make sense at all. What are some other scalar quantities? Temperature would be a scalar quantity. It's just how fast the molecules are moving, but it's not in one certain direction. Pressure would be another one that is scalar. It's not directional. It's not in one direction, the pressure is remember, pressure air pressure is the one that I always think of is going to be in all direction, so we wouldn't say that. Let's see, mass. The mass of something is going to be a scalar quantity as well so it it doesn't change. Now wait and we'll talk more about that later and would actually be a a vector quantity. let's see if I'm missing any. now I think this would be good so let's change color for a second. So, displacement is how far you move from a location and that's in a direction. So we call that a vector quantity acceleration I mentioned before. force is going to be a vector and will do these force diagrams which are really fun later in the year. Drag is something slowing you down, so if your car it's what's slowing you down in the opposite direction of your movement, so the direction is important. Momentum is a product of velocity in the mass of an object, and lift we get from like an airplane wing. That would be a vector quantity because it's in a direction. So these are all vector quantities, the ones that I circled in red, but there are way more that we're going to find out there. And scalar quantities remember it's simply just magnitude or how big it is. And so as we go through physics be thinking to yourself is this a scalar quantity or vector? And if it's vector, my problem is a little bit harder, but like Angry Birds it's more fun when you go the vector route. And so, I hope that's helpful and have a great day!

Typically the vectors used in meteorology and atmospheric science have two or three dimensions. Let’s think of two three-dimensional vectors of some variable (e.g., wind, force, momentum):

\[\vec{A}=\vec{i} A_{x}+\vec{j} A_{y}+\vec{k} A_{z}\]

\[\vec{B}=\vec{i} B_{x}+\vec{j} B_{y}+\vec{k} B_{z}\]

Sometimes we designate vectors with **bold lettering**, especially if the word processor does not allow for arrows in the text. When Equations [8.3] are written with vectors in bold, they are:

\[\mathbf{A}=\mathbf{i} A_{x}+\mathbf{j} A_{y}+\mathbf{k} A_{z}\]

\[\mathbf{B}=\mathbf{i} B_{x}+\mathbf{j} B_{y}+\mathbf{k} B_{z}\]

Be comfortable with both notations for representing vectors.

In the equations for vectors, * A_{x}* and

*are the magnitudes of the two vectors in the*

*B*_{x}*(east–west) direction, for which \(\vec{i}\) or i is the unit vector;*

*x**and*

*A*_{y}*are the magnitudes of the two vectors in the*

*B*_{y}*(north–south) direction, for which \(\vec{j}\) or*

*y***is the unit vector; and**

*j**and*

*A*_{z}*are the magnitudes of the two vectors in the*

*B*_{z}*(up–down) direction, for which \(\vec{k}\) or*

*z***is the unit vector. Unit vectors are sometimes called direction vectors.**

*k*Sometimes we want to know the magnitude (length) of a vector. For example, we may want to know the wind speed but not the wind direction. The magnitude of \(\vec{A}\) , or * A*, is given by:

\[|\vec{A}|=\sqrt{\left(A_{x}^{2}+A_{y}^{2}+A_{z}^{2}\right)}\]

We often need to know how two vectors relate to each other in atmospheric kinematics and dynamics. The two most common vector operations that allow us to find relationships between vectors are the **dot product** (also called the scalar product or inner product) and the **cross product** (also called the vector product).

The dot product of two vectors *A* and

*that have an angle \(β\) between them is given by:*

**B**\[ \begin{align} \vec{A} \cdot \vec{B} &= A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z} \\[4pt] &=|\vec{A}||\vec{B}| \cos \beta \label{eq5}\end{align}\]

We can identify two extremes

\[

\vec{A} \cdot \vec{B} = \begin{cases}

|\vec{A}||\vec{B}| & \text{if } \vec{A} \parallel \vec{B} \\

0 & \text{if } \vec{A} \perp \vec{B}

\end{cases}

\]

The dot product is simply the magnitude of one of the vectors, for example **A**, multiplied by the projection of the other vector,

**, onto**

*B***, which is just B cosβ**

*A***and**

*A**are parallel to each other, then their dot product is*

**B***. If they are perpendicular to each other, then their dot product is 0. The dot product is a scalar and therefore has magnitude but no direction.*

*AB*Also note that the unit vectors (a.k.a., direction vectors) have the following properties:

\[\vec{i} \cdot \vec{i}=\vec{j} \cdot \vec{j}=\vec{k} \cdot \vec{k}=1\]

\[\vec{i} \cdot \vec{j}=\vec{i} \cdot \vec{k}=\vec{j} \cdot \vec{k}=\vec{j} \cdot \vec{i}=\vec{k} \cdot \vec{i}=\vec{k} \cdot \vec{j}=0\]

\[\vec{i} \cdot \vec{A}=A_{x}\]

\[\vec{B} \cdot \vec{A}=\vec{A} \cdot \vec{B}\]

Note that the dot product of the unit vector with a vector simply selects the magnitude of the vector's component in that direction \(\left.\overrightarrow{(i} \cdot \vec{A}=A_{x}\right)\) and that the dot product is commutative \((\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A})\)

Equation \ref{eq5} can be rearranged to yield an expression for \(\cos β\) in terms of the vector components and vector magnitudes:

\[\cos \beta=\frac{A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}}{|\vec{A}||\vec{B}|}\]

The cross product of two vectors *A* and

*that have an angle ββ between them is given by:*

**B**\[\vec{A} \times \vec{B}=\left(\begin{array}{ccc}{\vec{i}} & {\vec{j}} & {\vec{k}} \\ {A_{x}} & {A_{y}} & {A_{z}} \\ {B_{x}} & {B_{y}} & {B_{z}}\end{array}\right)\]

\[\vec{A} \times \vec{B}=\left(A_{y} B_{z}-A_{z} B_{y}\right) \vec{i}-\left(A_{x} B_{z}-A_{z} B_{x}\right) \vec{j}+\left(A_{x} B_{y}-A_{y} B_{x}\right) \vec{k}\]

The magnitude of the cross product is given by:

\[|\vec{A} \times \vec{B}|=|\vec{A}||\vec{B}| \sin \beta\]

We can identify two extremes

\[

|\vec{A} \times \vec{B}| = \begin{cases}

0 & \text{if } \vec{A} \parallel \vec{B} \\

|\vec{A}||\vec{B}| & \text{if } \vec{A} \perp \vec{B}

\end{cases}

\]

where \(β\) is the angle between *A* and

*, with \(β\) increasing from*

**B***to*

**A***.*

**B**Note that the cross product is a vector. The direction of the cross product is at right angles to *A* and

*, in the right hand sense. That is, use the right hand rule (have your hand open, curl it from*

**B***to*

**A****, and**

*B**x*

**A***will be in the direction of your right thumb). The magnitude of the cross product can be visualized as the area of the parallelogram formed from the two vectors. The direction is perpendicular to the plane formed by vectors*

**B***and*

**A***. Thus, if*

**B***and*

**A***are parallel to each other, the magnitude of their cross product is 0. If*

**B***and*

*A**are perpendicular to each other, the magnitude of their cross product is*

*B**.*

*AB*The following video (2:06) reminds you about the right-hand rule for cross products.

**Right-hand Rule for Vector Cross Product**

**Click here for transcript of the Right-hand Rule for Vector Cross product**-
We're going to do a couple more examples of finding vector cross product. Suppose that I give you these two vectors a and B, which both lie in the plane of, look its my hands, which both lie in the plane of the page. Ok, so there are a and B. You want to find the direction of a cross B. To find the magnitude you do a times B times the sine of the angle between them, but we just want to find the direction right now, and to do this we're going to use the right hand rule, but first we can use a little bit of logic. So, first of all logic says this, whatever the direction of a cross B is which let's call that c, a cross b the we'll call that c. It has to be perpendicular to both a and B or perpendicular to the plane of the page. Well there are only two directions that that could be, right. What that means is that c either must point straight out of the page or it must point straight into the page. And, to figure out which one of those two directions it is, what we're going to have to do is we're gonna have to put our fingers along a. So there are two ways to do that. You can either put your fingers along a this way, or you could put your fingers along a this way, and you have to do it in the way that will let you swing a down into b like it was a little hinge. So, if you try that notice if you do it this way, yeah it's the wrong way right. You'd have to swing all the way the long way around. If you want to just simply fold a into b the way to do that is to put your fingers this way then you can curl them down this way. Notice when you do that your thumb is pointing into the page, so therefore, the answer is that c is into the page... and actually I got marker on my wall. Actually, the way we represent that is that's represented into the page is represented by a little X with a circle around it. You're supposed to think of it like the tail feathers of an arrow that's pointing into the page.

It follows that the cross products of the unit vectors are given by:

\[\vec{i} \times \vec{j}=\vec{k} \quad \vec{j} \times \vec{k}=\vec{i} \quad \vec{k} \times \vec{i}=\vec{j}\]

\[\vec{i} \times \vec{j}=-\vec{j} \times \vec{i}\]

Note finally that \[\vec{A} \times \vec{B}=-\vec{B} \times \vec{A}\]

We sometimes need to take derivatives of vectors in all directions. For that we can use a special vector derivative called the Del operator, \(\vec{\nabla}\)

Del is a vector differential operator that tells us the change in a variable in all three directions. Suppose that we set out temperature sensors on a mountain so that we get the temperature, * T*, as a function of

*,*

*x**, and*

*y**. Then \(\vec{\nabla}\)*

*z**would give us the change of*

*T**in the*

*T**,*

*x**, and*

*y**directions.*

*z*\[\vec{\nabla}=\vec{i} \frac{\partial}{\partial x}+\vec{j} \frac{\partial}{\partial y}+\vec{k} \frac{\partial}{\partial z}\]

The Del operator can be used like a vector in dot products and cross products but not in sums and differences. It does not commute with vectors and must be the partial derivative of some variable, either a scalar or a vector. For example, we can have the following with del and a vector *A*:

\(\vec{\nabla} \cdot \vec{A}=\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z},\) which is a scalar

\(\vec{\nabla} T=\vec{i} \frac{\partial T}{\partial x}+\vec{j} \frac{\partial T}{\partial y}+\vec{k} \frac{\partial T}{\partial z},\) which is a vector even though \(T\) is a scalar

\(\vec{A} \cdot \vec{\nabla} T=A_{x} \frac{\partial T}{\partial x}+A_{y} \frac{\partial T}{\partial y}+A_{z} \frac{\partial T}{\partial z},\) which is a scalar