4.3.3 Determine an orthonormal basis for the basis   eeee 1 , 2 , 3 , 4  for the vector space defined

M 41 4.3.3 Determine an orthonormal basis for the basis   eeee 1 , 2 , 3 , 4  for the vector space defined

e 1    , e 2   , e 3    , e 4   (4.3.55)

 2  3 i 

 5 i

Sec. 4.3 • Orthogonal Vectors and Orthonormal Bases

267

The correct answer is

32 4.3.4 You are given a matrix  A M defined by

A 2 3  (4.3.57)

Find an orthonormal basis for the image space RA  if A .

32 4.3.5 You are given a matrix  A M defined by

A 2 3  (4.3.58)

Find an orthonormal basis that spans RA  .

268

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Sec. 4.4 • Orthonormal Bases in Three Dimensions 269

Section 4.4. Orthonormal Bases in Three Dimensions

The real inner product space of dimension three is especially important in applied mathematics. It arises as the underlying mathematical structure of almost all application of the broad area known as applied mechanics. For this reason and others, in this section, we shall built upon the results of Section 4.3 and study this special case in greater detail. We begin with the assumption that we have an orthonormal basis for this real inner product space V that we will

denote by  iii 1 ,, 2 3  . Given any basis for V , the Gram-Schmidt process discussed in Section 4.3

allows this orthonormal basis to be constructed. It is customary to illustrate this basis with a figure like the following

The first topic we wish to discuss is the basis change from the orthonormal basis  iii 1 ,, 2 3 

to a second orthonormal basis ˆˆˆ  iii 1 ,, 2 3  . Geometrically, we can illustrate a possible second basis

on the above figure as follows:

Chap. 4 •

INNER PRODUCT SPACES

i 3 ˆi 3 ˆi 2

ˆi

As follows from (4.3.6), the fact that the two bases are orthonormal is reflected in the requirements

ii ˆˆ j , q   jq jk ,  1,2,3 (4.4.2)

As we have discussed several times and discussed in detail in Section 2.6, the two sets of bases are related by an expression of the form

3 ˆ i j   Q kj k i for j  1, 2,3 (4.4.3)

It follows from (4.4.3) and (4.4.1) that the coefficients of the transition matrix are given by

ii s , ˆ j  i s ,  Q kj k i   Q kj ii s , k  Q sj (4.4.4)

Because we are dealing with a real vector space, the result (4.4.4) combined with the definition (4.2.10) tells us that Q sj  i i is the cosine of the angle between the vectors s , ˆ j i s and ˆ i j . These

cosines are the usual direction cosines that are familiar from elementary geometry.

The coefficients of the transition matrix

Sec. 4.4 • Orthonormal Bases in Three Dimensions

 Q 11 Q 12 Q 13 

Q   Q    Q Q Q   kj 21 22 23 (4.4.5)

   Q 31 Q 32 Q 33  

are restricted by the two requirements (4.4.1) and (4.4.2). The nature of the restriction is revealed if we substitute (4.4.3) into (4.4.2) to obtain

ii ˆˆ j , q   Q sj s i ,  Q kq k i   QQ sj kq ii s , k   jq (4.4.6)

If we now utilize (4.4.1), (4.4.6) reduces to

 QQ sj kq  sk   QQ kj kq   jq (4.4.7)

The matrix form of (4.4.7) 2 is

T QQ  (4.4.8) I

Because the transition matrix is nonsingular, it follows from (4.4.8) that

 1 T Q  Q (4.4.9)

Therefore, the inverse of the transition matrix between two orthonormal bases is equal to its transpose . The special result (4.4.9) also implies

T QQ  (4.4.10) I

in addition to (4.4.8). Also, because of (1.10.21), it follows from (4.4.8) that

2 det 

QQ   det Q det Q   det Q   1 (4.4.11)

and, thus,

det Q   (4.4.12) 1

The transition matrix Q is an example of what is known as an orthogonal matrix. It has the property, as reflected in the construction above, of preserving lengths and angles.

If we view the transition matrix as consisting of 3 column vectors qq 1 , 2 and q 3 defined by

Chap. 4 •

INNER PRODUCT SPACES

for j  1,2,3 (4.4.13)

then it is easy to restate the orthogonality condition (4.4.8) as a condition of orthogonality on the column vectors (4.4.13). The formal condition that reflects this fact is

T qq

k   jk (4.4.14)

Example 4.4.1 : An elementary example of the basis change described above is one where Q takes the simple form

 Q 11 Q 12 0 

Q    Q kj    Q Q 0  (4.4.15)

With this choice for the transition matrix, the basis change defined by (4.4.3) reduces to

i ˆ 1  Q i 11 1  Q i 21 2

i ˆ 2  Q i 12 1  Q i 22 2 (4.4.16)

where, from (4.4.8)

Q 11  Q 21  1

Q 12  Q 22  1 (4.4.17)

QQ 11 12  QQ 21 22  0

and, from(4.4.10),

Q 11  Q 12  1

Q 21  Q 22  1 (4.4.18)

QQ 11 21  QQ 12 22  0

For the basis change defined by (4.4.16), the above figure is replaced by

Sec. 4.4 • Orthonormal Bases in Three Dimensions

Thus, the choice (4.4.16) corresponds to some type of a rotation about the 3 axis. If we view the above figure from the prospective of a rotation in the plane, the result is

ˆi i 2 2

ˆi 1

where the 3 axis can be viewed as pointing out of the page. The six equations (4.4.17) and (4.4.18)

have certain obvious implications. First, (4.4.17) 1 and (4.4.18) 1 imply

Q 21  Q 12 (4.4.19)

The result (4.4.19) reduce (4.4.17) 2 and (4.4.18) 2 to the same equation. Also, given (4.4.19), it follows from (4.4.17) 1 and (4.4.17) 2 that

Q 22  Q 11 (4.4.20)

Chap. 4 •

INNER PRODUCT SPACES

Given (4.4.19) and (4.4.20), equation (4.4.17) 3 or, equivalently, (4.4.18) 3 yield

QQ 11 21  QQ 12 22  Q 11   Q 12   Q 12   Q 11   0 (4.4.21)

If we exclude the trivial case where Q 11 or Q 12 are zero, (4.4.21) says that the multiplicity reflected

in (4.4.19) and (4.4.20) must be paired such that if

Q 22  Q 11 (4.4.22)

then

Q 21   Q 12 (4.4.23)

Given (4.4.22) and (4.4.23), we see that the first column of (4.4.15) determines the two possible choices for the second column of (4.4.15). The remaining condition on the elements of the first

column is (4.4.18) 1 , repeated,

Q 11  Q 12  (4.4.24) 1

or, equivalently,

12  1   Q 11 (4.4.25)

Equations (4.4.22) through (4.4.25) reduce the transition matrix (4.4.15) to eight possible choices. Four of these are as follows:

Case 1:

  Q

 21 Q 22 0  1  

Q 11  Q  11 0  (4.4.26)

Case 2:

Sec. 4.4 • Orthonormal Bases in Three Dimensions

11 1  Q 11 0

 Q 11 Q 12 0  

Q    Q kj    Q 21 Q 0 22   1  Q

Case 3:

11  1   Q 0 

 Q 11 Q 12 0  

kj   Q 21 Q 22 0   1   Q 11  Q 11 0   (4.4.28) 

and, finally,

Case 4:

11 1   Q 0 11 

The other four cases result from formally replacing Q 11 with  Q 11 in the above four. Returning to

the four cases listed, the details become a little more transparent if we introduce an angle  that

makes cos  the direction cosine between i 1 and ˆi 1 . Given this interpretation, we can use (4.4.4) to

write

Q 11  ii 1 , ˆ 1  cos  (4.4.30)

which reduces the above four cases to

Case 1:

 cos   sin  0 

Q  sin  cos  0   (4.4.31)

Case 2:

276

Chap. 4 •

INNER PRODUCT SPACES

 cos  sin  0  Q   sin  cos  0  (4.4.32)

Case 3:

 cos  sin  0  Q   sin   cos  0  (4.4.33)

and

Case 4:

 cos   sin  0  Q   sin   cos  0  (4.4.34)

The first two cases correspond to the situation where det Q  and the second two cases 1 correspond to the case where det Q   . Also, Case 2 differs from Case 1 and case 4 from case 3 1 by the sign of the angle. If Cases 1 and 3 represent some sort of rotation by an amount  , then Cases 2 and 4 represent the same type of rotation by an amount   . The following four figures

display geometrically these four cases:

Sec. 4.4 • Orthonormal Bases in Three Dimensions

In a sense that we will describe, Case 1 is the fundamental case illustrated by this example. Cases 2 and 3 are similar to 1 and 4, respectively. They simply represent rotations by a negative

angle. Case 3 can be thought of as being Case 2 followed by another basis change ˆ i 1  ˆ i 1 and

i ˆ 2  ˆ i 2 . Likewise Case 4 can be thought of as being Case 1 followed by another basis change

i ˆ 1  ˆ i 1 and ˆ i 2  i ˆ 2 . The bottom line of all of this is that Case 1 represents a basic rotation.

Cases 2,3 and 4 represent a negative rotation in one case and a rotation followed by a second basis change which simply flips one of the axes. Cases 3 and 4, as mentioned, have the property that

det Q   . Rotations with this property involve the kind of axis inversion illustrated by Cases 3 1 and 4. They are sometimes called improper rotations. If we rule out improper rotations and

recognize that Case 2 is a special case of Case 1, the transition matrix that characterizes a rotation about the i 3 axis is (4.4.31), repeated,

 cos   sin  0 

Q  sin  cos

Chap. 4 •

INNER PRODUCT SPACES

It should be evident that the transition matrix for rotations about the other axes are

for a rotation about the i 2 axis and

Q   0 cos   sin   (4.4.37)

  0 sin  cos   

for a rotation about the i 1 axis.

Exercises

4.4.1 In the applications it is often the case where the angle  depends upon a parameter t such as

the time. The result is, for example, that the orthogonal matrix Q depends upon t . Given Q  t defined by

 cos   t  sin   t 0 

Qt   sin   t cos   t 0  (4.4.38)

Show that

d   t 

 dt

dQ t   d   t

0 0 Qt  (4.4.39)

dt

dt

Sec. 4.4 • Orthonormal Bases in Three Dimensions

279

d   t 

dt

 d   t

d   t

Therefore, the skew symmetric matrix 

0 0  , which is determined by

 dt 

dt

determines the angular velocity for the rotation (4.4.35)

4.4.2 Generalize the results of Exercise 4.4.1 for an arbitrary orthogonal matrix and show that

dQ t 

 ZtQt  (4.4.40)

dt

where T Zt  is a skew symmetric matrix, i.e., where Zt   Zt  . The result (4.4.40)

generalizes the special result (4.4.39) and shows that the angular velocity of the basis ˆˆˆ  iii 1 ,, 2 3 

with respect to the basis  iii 1 ,, 2 3  is determined by Zt  . As a skew symmetric matrix in three

dimensions, Zt  can have only three nonzero components. It is customary to use these three components to define a three dimensional vector which is known as the angular velocity vector of

the basis  iii ˆˆˆ 1 ,, 2 3  with respect to the basis  iii 1 ,, 2 3  .

280

Chap. 4 •

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Sec. 4.5 • Euler Angles 281 Section 4.5 Euler Angles 3

Geometric constructions like the one discussed in Section 4.4 arise in a lot of applications. There is an entire branch of mechanics where one studies the motion of rigid bodies like, for

example, gyroscopes, where the basis  iii 1 ,, 2 3  is a reference or fixed orientation in space and the

basis ˆˆˆ  iii 1 ,, 2 3  is fixed to the rigid body and, thus, defines the position of the rigid body relative to the fixed orientation. Another application is when one thinks of the basis  iii ˆˆˆ 1 ,, 2 3  as fixed to the

body of an aircraft and its orientation relative to  iii 1 ,, 2 3  gives the orientation of the aircraft.

These applications get rather complicated as one tries to characterize the position of, for example, a rigid body as a consequence of a general rotation. The usual approach is to view the final position as the result of a sequence of three rotations of the form of (4.4.35) through (4.4.37). In aerodynamics the three rotations are known by the names of roll, pitch and yaw. In a more general

context of rigid body dynamics they are known as the Euler angles. 4

The usual way these three rotations are represented is shown in the following figure:

ˆi  2

ˆi 3

ˆi

The sequence of rotations are:

1. Rotate about the i 3 axis by an angle .

3 Information about Leonhard Paul Euler can be found at http://en.wikipedia.org/wiki/Leonhard_Euler . 4 There is an excellent discussion of Euler angles at http://en.wikipedia.org/wiki/Euler_angles .

Chap. 4 •

INNER PRODUCT SPACES

2. Rotate about the “rotated” i 1 axis by an angle  which aligns the “rotated” i 3 with ˆi 3 .

3. Rotate about the ˆi 3 axis by an angle  to align with ˆi 1 and ˆi 2

The details to construct these bases changes are complicated. We really need four bases to fully

characterize the three rotations listed. Two of them are the first basis,  iii 1 ,, 2 3  , and the final basis,