Published on *METEO 300: Fundamentals of Atmospheric Science* (https://www.e-education.psu.edu/meteo300)

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.

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):

$$\begin{array}{l}\overrightarrow{A}=\overrightarrow{i}{A}_{x}+\overrightarrow{j}{A}_{y}+\overrightarrow{k}{A}_{z}\hfill \\ \overrightarrow{B}=\overrightarrow{i}{B}_{x}+\overrightarrow{j}{B}_{y}+\overrightarrow{k}{B}_{z}\hfill \end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaauaadaqaceaaaeaaqqaaaaaaaaGySf2yRbWdbiqadgeapaGbaSaapeGaeyypa0JabmyAa8aagaWca8qacaWGbbWdamaaBaaaleaapeGaamiEaaWdaeqaaOWdbiabgUcaRiqadQgapaGbaSaapeGaamyqa8aadaWgaaWcbaWdbiaadMhaa8aabeaak8qacqGHRaWkceWGRbWdayaalaWdbiaadgeapaWaaSbaaSqaa8qacaWG6baapaqabaaakeaapeGabmOqa8aagaWca8qacqGH9aqpceWGPbWdayaalaWdbiaadkeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaey4kaSIabmOAa8aagaWca8qacaWGcbWdamaaBaaaleaapeGaamyEaaWdaeqaaOWdbiabgUcaRiqadUgapaGbaSaapeGaamOqa8aadaWgaaWcbaWdbiaadQhaa8aabeaaaaaaaa@51F4@

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:

$$\begin{array}{l}A=i{A}_{x}+j{A}_{y}+k{A}_{z}\hfill \\ B=i{B}_{x}+j{B}_{y}+k{B}_{z}\hfill \end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaauaadaqaceaaaeaaqqaaaaaaaaGySf2yRbWdbiaahgeacqGH9aqpcaWHPbGaamyqa8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qacqGHRaWkcaWHQbGaamyqa8aadaWgaaWcbaWdbiaadMhaa8aabeaak8qacqGHRaWkcaWHRbGaamyqa8aadaWgaaWcbaWdbiaadQhaa8aabeaaaOqaa8qacaWHcbGaeyypa0JaaCyAaiaadkeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaey4kaSIaaCOAaiaadkeapaWaaSbaaSqaa8qacaWG5baapaqabaGcpeGaey4kaSIaaC4AaiaadkeapaWaaSbaaSqaa8qacaWG6baapaqabaaaaaaa@508C@

Yet another notation uses parentheses and commas, as in ** A **= (

In the equations for vectors, *A _{x}* and

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 $\stackrel{\rightharpoonup}{A}$ , or *A*, is given by:

$\left|\overrightarrow{A}\right|=\sqrt{\left({A}_{x}^{2}+{A}_{y}^{2}+{A}_{z}^{2}\right)}$

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

$$\begin{array}{c}\overrightarrow{A}\u2022\overrightarrow{B}={A}_{x}{B}_{x}+{A}_{y}{B}_{y}+{A}_{z}{B}_{z}\\ =\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\mathrm{cos}\beta \\ =\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\text{if}\overrightarrow{A}\text{isparallelto}\overrightarrow{B}\\ =0\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\overrightarrow{A}\perp \overrightarrow{B}\end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@80D1@

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

Also note that the unit vectors have the following properties:

$$\begin{array}{l}\overrightarrow{i}\xb7\overrightarrow{i}=\overrightarrow{j}\xb7\overrightarrow{j}=\overrightarrow{k}\xb7\overrightarrow{k}=1\\ \overrightarrow{i}\xb7\overrightarrow{j}=\overrightarrow{i}\xb7\overrightarrow{k}=\overrightarrow{j}\xb7\overrightarrow{k}=\overrightarrow{j}\xb7\overrightarrow{i}=\overrightarrow{k}\xb7\overrightarrow{i}=\overrightarrow{k}\xb7\overrightarrow{j}=0\\ \overrightarrow{i}\xb7\overrightarrow{A}={A}_{x}\\ \overrightarrow{B}\xb7\overrightarrow{A}=\overrightarrow{A}\xb7\overrightarrow{B}\end{array}$$

Note that the dot product of the unit vector with a vector simply selects the magnitude of the vector's component in that direction ($\overrightarrow{i}\xb7\overrightarrow{A}={A}_{x}$) and that the dot product is commutative $(\overrightarrow{A}\xb7\overrightarrow{B}=\overrightarrow{B}\xb7\overrightarrow{A})$ .

Equation [8.4] can be rearranged to yield an expression for $\mathrm{cos}\beta $[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaci4yaiaac+gacaGGZbGaeqOSdigaaa@3ADE@ in terms of the vector components and vector magnitudes:

$$\mathrm{cos}\beta =\frac{{A}_{x}{B}_{x}+{A}_{y}{B}_{y}+{A}_{z}{B}_{z}}{\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaci4yaiaac+gacaGGZbGaeqOSdiMaeyypa0ZaaSaaa8aabaWdbiaadgeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaamOqa8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qacqGHRaWkcaWGbbWdamaaBaaaleaapeGaamyEaaWdaeqaaOWdbiaadkeapaWaaSbaaSqaa8qacaWG5baapaqabaGcpeGaey4kaSIaamyqa8aadaWgaaWcbaWdbiaadQhaa8aabeaak8qacaWGcbWdamaaBaaaleaapeGaamOEaaWdaeqaaaGcbaWdbmaaemaapaqaa8qaceWGbbWdayaalaaapeGaay5bSlaawIa7amaaemaapaqaa8qaceWGcbWdayaalaaapeGaay5bSlaawIa7aaaaaaa@539B@

The cross product of two vectors * A* and

$\begin{array}{l}\overrightarrow{A}\times \overrightarrow{B}=\left(\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ {A}_{x}& {A}_{y}& {A}_{z}\\ {B}_{x}& {B}_{y}& {B}_{z}\end{array}\right)\\ \\ \overrightarrow{A}\times \overrightarrow{B}=\left({A}_{y}{B}_{z}-{A}_{z}{B}_{y}\right)\overrightarrow{i}-\left({A}_{x}{B}_{z}-{A}_{z}{B}_{x}\right)\overrightarrow{j}+\left({A}_{x}{B}_{y}-{A}_{y}{B}_{x}\right)\overrightarrow{k}\end{array}$

The magnitude of the cross product is given by:

$$\begin{array}{l}\left|\overrightarrow{A}\times \overrightarrow{B}\right|=\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\mathrm{sin}\beta \hfill \\ \begin{array}{l}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0\text{if}\overrightarrow{A}\text{isparallelto}\overrightarrow{B}\\ \text{}=\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\text{if}\overrightarrow{A}\perp \overrightarrow{B}\end{array}\hfill \end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@836A@

where $\beta $ is the angle between * A* and

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

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

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

$$\begin{array}{l}\overrightarrow{i}\times \overrightarrow{j}=\overrightarrow{k}\text{\hspace{1em}}\text{\hspace{1em}}\overrightarrow{j}\times \overrightarrow{k}=\overrightarrow{i}\text{\hspace{1em}}\text{\hspace{1em}}\overrightarrow{k}\times \overrightarrow{i}=\overrightarrow{j}\\ \\ \overrightarrow{i}\times \overrightarrow{j}=-\overrightarrow{j}\times \overrightarrow{i}\end{array}$$

Note finally that $\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}$.

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

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*, *y*, and *z*. Then $\overrightarrow{\nabla}$*T* would give us the change of *T* in the *x*, *y*, and *z* directions.

$\overrightarrow{\nabla}=\overrightarrow{i}\frac{\partial}{\partial x}+\overrightarrow{j}\frac{\partial}{\partial y}+\overrightarrow{k}\frac{\partial}{\partial z}$

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 operator and a vector * A*:

$$\begin{array}{l}\stackrel{}{}\end{array}$$

$$\begin{array}{l}\overrightarrow{\nabla}\xb7\overrightarrow{A}=\frac{\partial {A}_{x}}{\partial x}+\frac{\partial {A}_{y}}{\partial y}+\frac{\partial {A}_{z}}{\partial z}\text{,whichisascalar}\\ \overrightarrow{\nabla}T=\overrightarrow{i}\frac{\partial T}{\partial x}+\overrightarrow{j}\frac{\partial T}{\partial y}+\overrightarrow{k}\frac{\partial T}{\partial z}\text{,whichisavectoreventhough}T\text{isascalar}\\ \overrightarrow{A}\xb7\overrightarrow{\nabla}T={A}_{x}\frac{\partial T}{\partial x}+{A}_{y}\frac{\partial T}{\partial y}+{A}_{z}\frac{\partial T}{\partial z}\text{,whichisascalar}\end{array}$$

- Find
**Practice Quiz 8-1**in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz. - When you feel you are ready, take
**Quiz 8-1**. You will be allowed to take this quiz only**once**. Good luck!