METEO 300
Fundamentals of Atmospheric Science

8.2 What you don’t know about vectors may surprise you!

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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.

Click for transcript of Scalars and Vectors

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):

A = i A x + j A y + k A z B = i B x + j B y + k B z MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@51F4@
[8.3]

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:

A=i A x +j A y +k A z B=i B x +j B y +k B z MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaauaadaqaceaaaeaaqqaaaaaaaaGySf2yRbWdbiaahgeacqGH9aqpcaWHPbGaamyqa8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qacqGHRaWkcaWHQbGaamyqa8aadaWgaaWcbaWdbiaadMhaa8aabeaak8qacqGHRaWkcaWHRbGaamyqa8aadaWgaaWcbaWdbiaadQhaa8aabeaaaOqaa8qacaWHcbGaeyypa0JaaCyAaiaadkeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaey4kaSIaaCOAaiaadkeapaWaaSbaaSqaa8qacaWG5baapaqabaGcpeGaey4kaSIaaC4AaiaadkeapaWaaSbaaSqaa8qacaWG6baapaqabaaaaaaa@508C@

Be comfortable with both notations.

In the equations for vectors, Ax and Bx are the magnitudes of the two vectors in the x (east–west) direction, for which i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. or iThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the unit vector; Ay and By are the magnitudes of the two vectors in the y (north–south) direction, for which j This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. or j is the unit vector; and Az and Bz are the magnitudes of the two vectors in the z (up–down) direction, for which k This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. or k is the unit vector. Unit vectors are sometimes called direction vectors.

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 A This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , or A, is given by:

| A |= ( A x 2 + A y 2 + A z 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

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 B that have an angle βThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. between them is given by:

A B = A x B x + A y B y + A z B z =| A || B |cosβ =| A || B | if  A  is parallel to  B =0if A B MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@80D1@
[8.4]

The dot product is simply the magnitude of one of the vectors, for example A, multiplied by the projection of the other vector, B, onto A, which is just B cosβThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.. Thus, if A and B are parallel to each other, then their dot product is AB. 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.

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

i · i = j · j = k · k =1 i · j = i · k = j · k = j · i = k · i = k · j =0 i · A = A x B · A = A · B This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.5]

Note that the dot product of the unit vector with a vector simply selects the magnitude of the vector's component in that direction ( i · A = A x ) and that the dot product is commutative ( A · B = B · A )This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. .

Equation [8.4] can be rearranged to yield an expression for cosβ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaci4yaiaac+gacaGGZbGaeqOSdigaaa@3ADE@ in terms of the vector components and vector magnitudes:

cosβ= A x B x + A y B y + A z B z | A || B | MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaabbaaaaaaaaIXwyJTgapeGaci4yaiaac+gacaGGZbGaeqOSdiMaeyypa0ZaaSaaa8aabaWdbiaadgeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaamOqa8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qacqGHRaWkcaWGbbWdamaaBaaaleaapeGaamyEaaWdaeqaaOWdbiaadkeapaWaaSbaaSqaa8qacaWG5baapaqabaGcpeGaey4kaSIaamyqa8aadaWgaaWcbaWdbiaadQhaa8aabeaak8qacaWGcbWdamaaBaaaleaapeGaamOEaaWdaeqaaaGcbaWdbmaaemaapaqaa8qaceWGbbWdayaalaaapeGaay5bSlaawIa7amaaemaapaqaa8qaceWGcbWdayaalaaapeGaay5bSlaawIa7aaaaaaa@539B@
[8.6]

The cross product of two vectors A and B that have an angle βThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. between them is given by:

A × B =( i j k A x A y A z B x B y B z ) A × B =( A y B z A z B y ) i ( A x B z A z B x ) j +( A x B y A y B x ) k This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.7]

The magnitude of the cross product is given by:

| A × B |=| A || B |sinβ =0        if  A  is parallel to  B          =| A || B | if  A B MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaerbdfwBIjxAHbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@836A@
[8.8]

where βThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the angle between A and B, with βThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. increasing from A to B.

Note that the cross product is a vector. The direction of the cross product is at right angles to A and B, in the right hand sense. That is, use the right hand rule (have your hand open, curl it from A to B, and A x B 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 A and B. Thus, if A and B are parallel to each other, the magnitude of their cross product is 0. If A and B are perpendicular to each other, the magnitude of their cross product is AB.

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

Click for Transcript of Right-hand Rule for Vector Cross Product

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

i × j = k j × k = i k × i = j i × j = j × i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.9]

Note finally that A × B = B × 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, This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. .

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 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.T would give us the change of T in the x, y, and z directions.

= i x + j y + k z This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.10]

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:

  This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
· A = A x x + A y y + A z z ,  which is a scalar T= i T x + j T y + k T z , which is a vector even though T is a scalar A · T= A x T x + A y T y + A z T z , which is a scalar This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

Quiz 8-1: Partial derivatives and vector operations.

  1. 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.
  2. When you feel you are ready, take Quiz 8-1. You will be allowed to take this quiz only once. Good luck!