Editing Euclidean vector (section)

===Conversion between multiple Cartesian bases===
All examples thus far have dealt with vectors expressed in terms of the same basis, namely, the ''e'' basis {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>}. However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector.  In the ''e'' basis, a vector '''a''' is expressed, by definition, as

<math display=block>\mathbf{a} = p\mathbf{e}_1 + q\mathbf{e}_2 + r\mathbf{e}_3.</math>

The scalar components in the ''e'' basis are, by definition,

<math display=block>\begin{align}
p &= \mathbf{a}\cdot\mathbf{e}_1, \\
q &= \mathbf{a}\cdot\mathbf{e}_2, \\
r &= \mathbf{a}\cdot\mathbf{e}_3.
\end{align}</math>

In another orthonormal basis ''n'' = {'''n'''<sub>1</sub>, '''n'''<sub>2</sub>, '''n'''<sub>3</sub>} that is not necessarily aligned with ''e'', the vector '''a''' is expressed as

<math display=block>\mathbf{a} = u\mathbf{n}_1 + v\mathbf{n}_2 + w\mathbf{n}_3</math>

and the scalar components in the ''n'' basis are, by definition,

<math display=block>\begin{align}
u &= \mathbf{a}\cdot\mathbf{n}_1, \\
v &= \mathbf{a}\cdot\mathbf{n}_2, \\
w &= \mathbf{a}\cdot\mathbf{n}_3.
\end{align}</math>

The values of ''p'', ''q'', ''r'', and ''u'', ''v'', ''w'' relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vector '''a''' in both cases.  It is common to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle).  In such a case it is necessary to develop a method to convert between bases so the basic vector operations such as addition and subtraction can be performed.  One way to express ''u'', ''v'', ''w'' in terms of ''p'', ''q'', ''r'' is to use column matrices along with a [[direction cosine matrix]] containing the information that relates the two bases.  Such an expression can be formed by substitution of the above equations to form

<math display=block>\begin{align}
u &= (p\mathbf{e}_1 + q\mathbf{e}_2 + r\mathbf{e}_3)\cdot\mathbf{n}_1, \\
v &= (p\mathbf{e}_1 + q\mathbf{e}_2 + r\mathbf{e}_3)\cdot\mathbf{n}_2, \\
w &= (p\mathbf{e}_1 + q\mathbf{e}_2 + r\mathbf{e}_3)\cdot\mathbf{n}_3.
\end{align}</math>

Distributing the dot-multiplication gives

<math display=block>\begin{align}
u &= p\mathbf{e}_1\cdot\mathbf{n}_1 + q\mathbf{e}_2\cdot\mathbf{n}_1 + r\mathbf{e}_3\cdot\mathbf{n}_1, \\
v &= p\mathbf{e}_1\cdot\mathbf{n}_2 + q\mathbf{e}_2\cdot\mathbf{n}_2 + r\mathbf{e}_3\cdot\mathbf{n}_2, \\
w &= p\mathbf{e}_1\cdot\mathbf{n}_3 + q\mathbf{e}_2\cdot\mathbf{n}_3 + r\mathbf{e}_3\cdot\mathbf{n}_3.
\end{align}</math>

Replacing each dot product with a unique scalar gives

<math display=block>\begin{align}
u &= c_{11}p + c_{12}q + c_{13}r, \\
v &= c_{21}p + c_{22}q + c_{23}r, \\
w &= c_{31}p + c_{32}q + c_{33}r,
\end{align}</math>

and these equations can be expressed as the single matrix equation

<math display=block>\begin{bmatrix} u \\ v \\ w \\ \end{bmatrix} = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} \begin{bmatrix} p \\ q \\ r \end{bmatrix}.</math>

This matrix equation relates the scalar components of '''a''' in the ''n'' basis (''u'',''v'', and ''w'') with those in the ''e'' basis (''p'', ''q'', and ''r'').  Each matrix element ''c''<sub>''jk''</sub> is the [[Direction cosine#Cartesian coordinates|direction cosine]] relating '''n'''<sub>''j''</sub> to '''e'''<sub>''k''</sub>.<ref name="dynon16">{{harvnb|Kane|Levinson|1996|pp=20–22}}</ref> The term ''direction cosine'' refers to the [[cosine]] of the angle between two unit vectors, which is also equal to their [[#Dot product|dot product]].<ref name="dynon16"/> Therefore,

<math display=block>\begin{align}
c_{11} &= \mathbf{n}_1\cdot\mathbf{e}_1 \\
c_{12} &= \mathbf{n}_1\cdot\mathbf{e}_2 \\
c_{13} &= \mathbf{n}_1\cdot\mathbf{e}_3 \\
c_{21} &= \mathbf{n}_2\cdot\mathbf{e}_1 \\
c_{22} &= \mathbf{n}_2\cdot\mathbf{e}_2 \\
c_{23} &= \mathbf{n}_2\cdot\mathbf{e}_3 \\
c_{31} &= \mathbf{n}_3\cdot\mathbf{e}_1 \\
c_{32} &= \mathbf{n}_3\cdot\mathbf{e}_2 \\
c_{33} &= \mathbf{n}_3\cdot\mathbf{e}_3
\end{align}</math>

By referring collectively to '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> as the ''e'' basis and to '''n'''<sub>1</sub>, '''n'''<sub>2</sub>, '''n'''<sub>3</sub> as the ''n'' basis, the matrix containing all the ''c''<sub>''jk''</sub> is known as the "[[transformation matrix]] from ''e'' to ''n''", or the "[[rotation matrix]] from ''e'' to ''n''" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from ''e'' to ''n''"<ref name="dynon16"/> (because it contains direction cosines).  The properties of a rotation matrix are such that its [[matrix inverse|inverse]] is equal to its [[matrix transpose|transpose]]. This means that the "rotation matrix from ''e'' to ''n''" is the transpose of "rotation matrix from ''n'' to ''e''".

The properties of a direction cosine matrix, C are:<ref>{{Cite book|title=Applied mathematics in integrated navigation systems|last=Rogers |first=Robert M. |date=2007|publisher=American Institute of Aeronautics and Astronautics|isbn=9781563479274|edition=3rd|location=Reston, Va.|oclc=652389481}}</ref>
* the determinant is unity, |C| = 1;
* the inverse is equal to the transpose;
* the rows and columns are orthogonal unit vectors, therefore their dot products are zero.

The advantage of this method is that a direction cosine matrix can usually be obtained independently by using [[Euler angles]] or a [[quaternion]] to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above.

By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.<ref name="dynon16"/>