Vector symbols explained

Here you will find the mathematical notation used for the n-vector page.

The notation system used for the n-vector page and in the files for download is presented in Chapter 2 of the following thesis: Gade (2018): Inertial Navigation - Theory and Applications. A simplified presentation is given here.

Coordinate frame

coordinate frame has a position (origin), and three axes (basis vectors) xy and z (orthonormal). Thus, a coordinate frame can represent both position and orientation, i.e. 6 degrees of freedom. It can be used to represent a rigid body, such as a vehicle or the Earth, and it can also be used to represent a "virtual" coordinate frame such as North-East-Down.

Coordinate frames are designated with capital letters, e.g. the three generic coordinate frames AB, and C.

We also have specific names for some common coordinate frames:

Coordinate frame  Description
E Earth
N North-East-Down
B Body, i.e. the vehicle

 

Note that it is no problem to only use the position or the orientation of a coordinate frame. E.g., in some cases, we just care about the position of point B and C (and sometimes we only care about the orientation of N).

General vector

A 3D vector given with numbers is written e.g. 
2 4 6 [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaWaamWaaeaafa qabeWabaaabaGaaGOmaaqaaiaaisdaaeaacaaI2aaaaaGaay5waiaa [email protected]@
. The three numbers are the vector components along the x-y- and z-axes of a coordinate frame. If the name of the vector is k, and the coordinate frame is A, we will use bold k and A as trailing superscript, i.e.:
 
k A = 2 4 6 [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa aaleqabaGaamyqaaaakiabg2da9maadmaabaqbaeqabmqaaaqaaiaa [email protected]@
 
Thus 
k A [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa [email protected]@
 is the 3D vector that is constructed by going 2 units along the x-axis of coordinate frame A, 4 units along the y-axis, and 6 along the z-axis. We say that the vector k is decomposed in A.
 

Position vector

Instead of the general vector k, we can have a specific vector that goes from A to B. This vector can be decomposed in CAB, and C are three arbitrary coordinate frames. We would write this vector:

p AB C [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa [email protected]@

In program code: p_AB_C

The letter p is used since this is a position vector (the position of B relative to A, decomposed/resolved in the axes of C).

Example a)

p EB E = 0 0 6371 km [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa aaleaacaWGfbGaamOqaaqaaiaadweaaaGccqGH9aqpdaWadaqaauaa beqadeaaaeaacaaIWaaabaGaaGimaaqaaiaaiAdacaaIZaGaaG4nai [email protected]@

From the subscript, we see that this is the vector that goes from E (center of the Earth) to B (the vehicle). The superscript tells us that it is decomposed in E, which we now assume has its z-axis pointing towards the North Pole. From the values, we see that the vector goes 6371 km towards the North Pole, and zero in the x and y directions. If we assume that the Earth is a sphere with radius 6371 km, we see that B is at the North Pole.

Example b)

p BC N = 50 60 5 m [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa aaleaacaWGcbGaam4qaaqaaiaad6eaaaGccqGH9aqpdaWadaqaauaa beqadeaaaeaacaaI1aGaaGimaaqaaiaaiAdacaaIWaaabaGaeyOeI0 [email protected]@

The vector goes from B, e.g. an aircraft, to C, e.g. an object. The vector is decomposed in N (which has North-East-Down axes). This means that C is 50 m north of B and 60 m east, and C is also 5 m above B.

Properties of the position vector

For the general position vector 
p AB C [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa [email protected]@
, we have the property:

p AB C = p BA C [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa aaleaacaWGbbGaamOqaaqaaiaadoeaaaGccqGH9aqpcqGHsislcaWH [email protected]@

I.e. swapping the coordinate frames in the subscript gives a vector that goes in the opposite direction. We also have:

p AD C = p AB C + p BD C [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa aaleaacaWGbbGaamiraaqaaiaadoeaaaGccqGH9aqpcaWHWbWaa0ba aSqaaiaadgeacaWGcbaabaGaam4qaaaakiabgUcaRiaahchadaqhaa [email protected]@

 

Math equation image

I.e., going from A to D is the same as first going from A to B, then from B to D. From the equation, we see that B is cancelled out. ABC, and D are arbitrary coordinate frames.

Rotation matrix

If we return to the general vector 
k A [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa [email protected]@
, we could also have a coordinate frame B, with different orientation than A. The same vector k could be expressed by components along the xy and z-axes of B instead A, i.e. it can also be decomposed in B, written 
k B [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa [email protected]@
. Note that the length of 
k B [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa [email protected]@
 equals the length of 
k A [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa [email protected]@
. We will now have the relation:

k A = R AB k B [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa aaleqabaGaamyqaaaakiabg2da9iaahkfadaWgaaWcbaGaamyqaiaa [email protected]@

R AB [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOuamaaBa [email protected]@
is the 9 element (3x3) rotation matrix (also called direction cosine matrix) that transforms vectors decomposed in B to vectors decomposed in A. Note that the B in 
R AB [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOuamaaBa [email protected]@
should be closest to the vector decomposed in B (following the "the rule of closest frames", see Section 2.5.3 in Inertial Navigation - Theory and Applications for details). If we need to go in the other direction, we have:
 
k B = R BA k A [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaC4AamaaCa aaleqabaGaamOqaaaakiabg2da9iaahkfadaWgaaWcbaGaamOqaiaa [email protected]@
 
Now we see that A is closest to A.

 

Properties of the rotation matrix

We have that

R AB = R BA T [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOuamaaBa aaleaacaWGbbGaamOqaaqabaGccqGH9aqpdaqadaqaaiaahkfadaWg aaWcbaGaamOqaiaadgeaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabe [email protected]@

where the T means matrix transpose. We also have the following property (closest frames are cancelled):

R AC = R AB R BC [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOuamaaBa aaleaacaWGbbGaam4qaaqabaGccqGH9aqpcaWHsbWaaSbaaSqaaiaa dgeacaWGcbaabeaakiaahkfadaWgaaWcbaGaamOqaiaadoeaaeqaaa [email protected]@

If we compare these properties with the position vector, we see that they are very similar: minus is replaced by transpose, and plus is replaced by matrix multiplication. AB, and C are three arbitrary coordinate frames.

n-vector

The n-vector is in almost all cases decomposed in E, and in the simplest form, we will write it

n E [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOBamaaCa [email protected]@

This simple form can be used in cases where there is no doubt about what the n-vector expresses the position of. In such cases, we can also express the position using e.g. the variables lat and long, without further specification.

However, if we are interested in the position of multiple objects, e.g. A and B, we must specify which of the two, both for n-vector and for latitude/longitude. In this case we will write:

n EA E [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOBamaaDa [email protected]@
and 
n EB E [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOBamaaDa [email protected]@
(program code: n_EA_E and n_EB_E

 

And

la t EA ,lon g EA [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamiBaiaadg gacaWG0bWaaSbaaSqaaiaadweacaWGbbaabeaakiaacYcacaWGSbGa [email protected]@
and 
la t EB ,lon g EB [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamiBaiaadg gacaWG0bWaaSbaaSqaaiaadweacaWGbbaabeaakiaacYcacaWGSbGa [email protected]@
 
The subscript E might seem redundant here, it could be sufficient to use only A or B. However, we have chosen to also include the E, since both n-vector and latitude/longitude are depending on the reference ellipsoid that is associated with E (see Section 4.1. in Gade (2010) for more about this). Note however, that the subscript rules (swapping and canceling) we had for 
p AB C [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa [email protected]@
 and 
R AB [email protected]@[email protected]@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOuamaaBa [email protected]@
 cannot be used for n-vector or lat/long.
 
For spherical Earth, we have a simple relation between 
p EB E [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa [email protected]@
 and 
n EB E [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCOBamaaDa [email protected]@
:

p EB E = n EB E r Earth + h EB = n EB E r Earth z EB [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaaCiCamaaDa aaleaacaWGfbGaamOqaaqaaiaadweaaaGccqGH9aqpcaWHUbWaa0ba aSqaaiaadweacaWGcbaabaGaamyraaaakiabgwSixpaabmaabaGaam OCamaaBaaaleaacaWGfbGaamyyaiaadkhacaWG0bGaamiAaaqabaGc cqGHRaWkcaWGObWaaSbaaSqaaiaadweacaWGcbaabeaaaOGaayjkai aawMcaaiabg2da9iaah6gadaqhaaWcbaGaamyraiaadkeaaeaacaWG fbaaaOGaeyyXIC9aaeWaaeaacaWGYbWaaSbaaSqaaiaadweacaWGHb GaamOCaiaadshacaWGObaabeaakiabgkHiTiaadQhadaWgaaWcbaGa [email protected]@

where 
r Earth [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamOCamaaBa [email protected]@
 is the radius of the Earth, 
h EB [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamiAamaaBa [email protected]@
is the height of B and 
z EB [email protected]@[email protected]@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamOEamaaBa [email protected]@
 is the depth. For more information about how to use n-vector in various calculations, see the 10 examples and Gade (2010).