CS 418: Homework #1
Assigned: Tuesday August 31, 2004
Due: Tuesday September 7, 2004 at the end of class
This homework is meant to help you assess your knowledge of linear algebra and geometry. We expect that
most of this should be review of material from your earlier courses. If you need a bit of a refresher for linear
algebra and geometry, most of what you need to know can be found in the textbook by Edward Angel,
specifically in the Appendices.
Please be organized when writing your answers to these questions. Make sure that all solutions are clearly
indicated and labelled with the question they are answering. Remember to write clearly and legibly. Un-
readable answers will receive 0 credit.
1. (20pt)
(a) Let v= [ 5 3 7 ]. Compute kvk, the vector length of v.
(b) Let u= [ 2 3 1 ] and v= [ 5 3 7 ]. Compute the angle umakes with v. Express your answer in
radians.
(c) Let u= [ 2 3 1 ] and v= [ 5 3 7 ]. Find a unit vector perpendicular to both uand v.
(d) Given a vector v= [x y], show that the vector u= [−y x] is orthogonal (i.e., perpendicular) to v.
2. (30pt) Suppose you are given matrices
M=
121
034
0−4 3
,N=
100
110
111
and vectors u=
2
3
1
, v =
x
y
z
.
(a) Compute the determinant detM.
(b) Compute the inverse M−1.
(c) Compute matrix-matrix multiplication MN.
(d) Compute matrix-vector multiplication Mv.
(e) Compute vector-matrix-vector multiplication uTMv.
(f) Compute the multiplication uvT.
3. (15pt) Suppose you are given the 2-D points p1=x0
y0and p2=x1
y1. Show that the equation of the
line between them is (y1−y0)x−(x1−x0)y=y1x0−x1y0.
4. (15pt) The equation of a plane in 3-D space is ax +by +cz +d= 0. Given three points p1,p2,p3find
the coefficients a, b, c for the plane containing the three given points.
5. (20pt) Matrix M(θ) is defined as follows:
M(θ) = cos θ−sin θ
sin θcos θ
