607
10/14/2011
Howell
TAKE MA/PH HOME PORTION OF
TEST II
(40 points)
GENERAL NOTES:
Due at the beginning of class on Friday, October 21.
This is NOT to be a team effort. Do your own work! (You may, of course, consult
with the instructor.)
Hand in the next two pages containing your final answers along with additional
sheets clearly showing how you got your answers (I will be looking at your work as
well as your final answers!)
Do not use decimal approximations. # "%"%#" ! Besides, using decimal
approximations will just make the arithematic harder in these problems.
For the most part, you might as well do the calculations by hand, especially since I
do not want to see any decimal approximations. However, at one point you may
have to multiply three $$ matrices together. You can use a computer math
system (Maple, Mathematica, etc.) for this. Just include a printout of your
program for the computation. (And remember, no decimal approximations.)
You can simplify life by remembering a little algebra, such as illustrated here:
"
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#!
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NAME:
MA/PH 607 Take Home Portion of Test II
I. The following concerns vectors in a traditional, threedimensional vector space. For assume
T these
problems, ei j kf
is a standard basis (i.e., T is orthonormal and right-handed), and let a and 9 be, respectively,
the vector1 and angle
a
#i
"j
#k
and
9
1
'
.
1. Find a right-handed, orthonormal basis U e b" b# b$ f such that b$ points in the
same direction as a . The b5 s should be expressed in terms of i , j and k (as was a ,
above).2 Be sure to convince me that U is both orthnormal and right-handed in your work.3
(8 pts.)
2. Find the unitary matrix U , such that
l v U U l v T for each vector v .
(6 pts.)
1The
vector a will be an axis vector.
2Hint:
The numbers may come out nicer if either b" or b# has only two nonzero components with
respect to T .
3Much
of the convincing can be in how you derive the basis vectors.
MA/PH 607 Take Home Portion of Test II
3. Let e be the linear transformation that
rotates each vector through angle 9
about the vector a . (I.e., this is the
rotation of space that rotates b" by an
angle of 9 towards b# .)
(See the figure, in which w e v . )
Reminder: T , U , a and 9 are
as on the previous page!
i. Find RU , the matrix of e with respect to U .
(9 pts.)
ii. Find RT , the matrix of e with respect to T .
(12 pts.)
iii. Let w e % i
(5 pts.)
!j
# k . Find l w T .
NAME: