INTRODUCTION TO LINEAR ALGEBRA, Second Edition
by Gilbert Strang
SOLUTIONS TO SELECTED EXERCISES
Christopher Heil
Spring 2000
CHAPTER 1
Introduction to Vectors
1.2 #13. Find two vectors
v
and
w
that are perpendicular to (1
,
1
,
1) and to each other.
Solution
There are many ways to go about this.
One way would be to write
v
= (
v
1
, v
2
, v
3
) and
w
= (
w
1
, w
2
, w
3
) and then to write down the equations that
v
and
w
must satisfy. These are:
v
·
(1
,
1
,
1) = 0,
w
·
(1
,
1
,
1) = 0, and
v
·
w
= 0. This gives a system of equations with the
v
i
and
w
i
as unknowns, that you could then try to solve to find the set of all possible
v
and
w
satisfying these requirements.
On the other hand, the problem just asks you to find one specific choice of
v
and
w
, not all
possible choices. So I think it is easiest to proceed first by inspection:
v
= (1
,
0
,
−
1) is clearly
perpendicular to (1
,
1
,
1) since their dot product is
v
·
(1
,
1
,
1) = 1
·
1 + 0
·
1
−
1
·
1 = 0. So,
we just have to find a
w
= (
w
1
, w
2
, w
3
) that is perpendicular to both of these vectors. This
w
must satisfy
w
·
(1
,
1
,
1) =
w
1
+
w
2
+
w
3
= 0
and
v
·
w
= (1
,
0
,
−
1)
·
w
=
w
1
−
w
3
= 0
.
There are infinitely many solutions to this system of equations; one particular solution is
w
= (1
,
−
2
,
1). There are many others.
1