# 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)

*x*,

*y*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 *A*, *B*, 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

*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.:

*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 *C*. *A*, *B*, and *C* are three arbitrary coordinate frames. We would write this vector:

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)**

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)**

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**

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

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. *A*, *B*, *C*, and *D* are arbitrary coordinate frames.

## Rotation matrix

*B*, with different orientation than

*A*. The same vector

*k*could be expressed by components along the

*x*,

*y*and

*z*-axes of

*B*instead

*A*, i.e. it can also be decomposed in

*B*, written

*B*to vectors decomposed in

*A*. Note that the

*B*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:

*A*is closest to

*A*.

**Properties of the rotation matrix**

We have that

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

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. *A*, *B*, 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

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`

and `n_EB_E`

)

And

*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

*n*-vector or lat/long.

*B*and

*n*-vector in various calculations, see the 10 examples and Gade (2010).