Exercises: Eigenvalues and Eigenvectors
1-8 • Find the eigenvalues of the given matrix.
~
[4
@~
13. Find the eigenvalues of the matrix
and find one eigenvector for each eigenvalue.
~l
~[~~~l
J(!)
[-i ~
6 -6 -6
009
L
~
~n
J@ [0-4
I1
-5
1
8. [~ ~ ~
4722
~l
~l
14. The)' = 2 eigenspace for the matrix
is two-dimensional. Find a basis for this eigenspace.
15. The)' = 1 eigenspace for the matrix
Find one eigenvector for the given matrix corresponding to
eigenvalue.
is two-dimensional. Find a basis for this eigenspace.
~
-
J@Are
------_-----------------------------
the following quantities vectors or scalars? Explain.
(a) The cost of a theater ticket
(b) The current in a river
(c) The initial flight path from Houston to Dallas
(d) The population of the world
j 0What is the relationship between the point (4, 7) and the
vector (4, 7)? Illustrate with a sketch.
J(}l Name all the equal vectors in the parallelogram shown.
,=-
9. A( -1,3), B(2,2)
OTI A(O, 3, I),
10. A(2,I), B(O, 6)
B(2, 3, -I)
12. A(4,0, -2),
8(4,2, I)
J9Find
the sum of the given vectors and illustrate
geometrically.
13. (-1,4),
(6, -2)
14. (-2,-I),
(5,7)
IS. (0,1,2),
(O,O, -3)
16. (-LO,2),
(0,4,0)
"B
4. Write each combination of vectors as a single vector,
-?
(a) PQ
-?
-?
+ QR
-?
(b) RP
-?
(c) QS - PS
-?
(d) RS
-?
+ PS
-?
-?
+ SP + PQ
21. -3i
R0s
~Find
fw
positive x-axis and I v I
but has
= 4,
find v in component form.
26. If a child pulls a sled through the snow on a level path with a
force of 50 N exerted at an angle of 38° above the horizontal,
find the horizontal and vertical components of the force.
27. A quarterback throws a football with angle of elevation 40° and
speed 60 ft/s. Find the horizontal and vertical components of
the velocity vector.
.-vG) Copy the vectors in the figure and use them to draw the
(b) a - b
(d)-~b
(f) b - 3a
a vector that has the same direction as (-2,4,2)
length 6.
~ If v lies in the first quadrant and makes an angle 7T/3 with the
/_
following vectors.
(a) a + b
(c) 2a
(e) 2a + b
22. (-4,2,4)
rID 8i - j + 4k
J@COpy the vectors in the figure and use them to draw the
following vectors.
(a) u + v
(b) u - v
(c) v + w
(d) w + v + u
~~
+ 7j
28-29 Find the magnitude of the resultant force and the angle it
makes with the positive x-axis,
28.
/20lli
As°
o 30°
29.
y
y
b
300N
x
x
161b
,,-,,--"""----,,--_-,,--------,-,--,-_.,_._--_._""""".._-
.....
7-12 Find-')a vector -')
a with representation given by the directed line
segment AB. Draw AB and the equivalent representation starting at
the origin.
7. A(2,3), B(-2,1)
8. A( - 2, -2), B(5,3)
30. The magnitude of a velocity vector is called speed. Suppose
that a wind is blowing from the direction N45°W at a speed of
50 km/h. (This means that the direction from which the wind
blows is 45° west of the northerly direction.) A pilot is steering
the work done by a constant force F is the dot product F • 0, where 0 is the displacement vector.
TIlUS
Ao
cz::J-~--'
EXAMPLE 7 A wagon is pulled a distance of 100 m along a horizontal path by a constant
force of 70 N. The handle of the wagon is held at an angle of 35° above the horizontal,
Find the work done by the force.
0:=0
If F and 0 are the force and displacement vectors, as pictured in Figure 7, then
the work done is
SOLUTION
L
°
I II I cos 35°
W =F .0 = F
D
FIGURE 7
=
•
(70)(100) cos 35° "" 5734 N'm = 5734 J
EXAMPLE 8 A force is given by a vector F = 3i + 4j + 5k and moves a particle from
the point P(2, 1,0) to the point Q(4,6, 2). Find the work done.
~
SOLUTION
The displacement vector is D = PQ = (2,5,2), so by Equation 12, the work
done is
W = F . D = (3,4,5)
. (2,5,2)
= 6 + 20 + 10 = 36
If the unit of length is meters and the magnitude of the force is measured in newtons,
then the work done is 36 joules.
•
~--------------I. Which of the following expressions are meaningful?Which are
J
meaningless?Explain.
(a) (a- b) . c
(b) (a . b)c
(c) I a I (b . c)
(d) a • (b
(e)a'b+c
(f)lal·(b+c)
11-12 If u is a unit vector, find u . v and u . w.
12.
u
+ c)
(yFind the dot product of two vectors if their lengths are 6
and ~and the angle between them is n/4.
w
~--II--~
w
leFinda'b.
3. a = (-2,j),
b = (-5,12)
4. a = (-2,3),
b = (0.7,1.2)
5. a
=
(4, 1, ~), b
=
(6, -3, -8)
6. a
=
(s, 2s, 3s),
b
=
7. a=i-2j
+3k,
13. (a) Show that i ' j = j , k = k ' i = O.
(b) Show that i .i = j . j = k . k = 1.
14. A street vendor sells a hamburgers,b hot dogs, and c soft
drinks on a givenday. He charges $2 for a hamburger,$[.50
for a hot dog, and $1 for a soft drink. If A = (a, b, c) and
/..
p = (2, 1.5, 1), what is the meaning of the dot productA . P?
(t, -t, 5t)
b=5i+9k
8. a = 4j - 3k, b = 2i + 4j + 6k
9. I a I = 6,
I b I = 5,
10. !a! = 3, !b] =
16,
the angle between a and b is 2n/3
~Find
the angle between the vectors. (First find an exact
expressionand then approximateto the nearest degree.)
15. a= (-8,6),
the angle between a and b is 45°
--_
...._ ..._ ....._
16. a ==
iJ3',
I),
b=(/f,3)
b == (0,5)
17.a=(3,-1,5},
18. a = (4,0,2),
b=(-2,4,3}
39. a
= 2i
b = (2, -I, o)
40. a
=i
[I?J a = j + k, b = i + 2j
20. a = i + 2j - 2k,
22. D(O, I, I), E( -2,4,3),
23. (a) a = (-5,3, 7), b = (6, -8,2)
(b)a= (4,6),
b= (-3,2)
(c) a = -i + 2j + 5k, b = 3i + 4j - k
(d)a=2i+6j-4k,
b=-3i-9j+6k
=
(4, - 12, -8)
=
2i - j
v
-I), find a vector b such that comp, b = 2.
44. Suppose that a and b are nonzero vectors.
(a) Under what circumstances is comp. b = compj, a?
(b) Under what circumstances is proj. b = proj b a?
F(I,2, - I)
whether the given vectors are orthogonal,
+ 2k,
+k
42. For the vectors in Exercise 36, find orth, b and illustrate by
drawing the vectors a, b, pro], b, and orth, b.
I1TI If a = (3,0,
parallel, or neither.
(c)u=(a,b,c),
b = i- j
+ ik
b - proj, b is orthogonal to a.
(It is called an orthogonal projection of b.)
21. A(I, 0), B(3, 6), c(- [,4)
6), v
+ j + k,
b= j
I!IJ Show that the vector orth, b =
b = 4i - 3k
21-22 Find, correct to the nearest degree, the three angles of the
triangle with the given vertices.
24. (a) u = (-3,9,
(b) u = i - j
+ 4k,
- 3k
------"-------------
j 90etermine
- j
46. A tow truck drags a stalled car along a road. The chain makes
an angle of 30° with the road and the tension in the chain is
1500 N. How much work is done by the truck in pulling the
car Ikm?
+k
v=(-b,a,O)
2S. "Use vectors to decide whether the triangle with vertices
P(l, - 3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled.
26. For what values of b are the vectors ( -6, b, 2) and (b, b2, b)
orthogonal?
Find a unit vector that is orthogonal to both i
4S. Find the work done by a force F = 8 i - 6 j + 9 k that moves
an object from the point (0, 10,8) to the point (6, 12,20) along
a straight line. The distance is measured in meters and the force
in newtons.
+ j and i + k.
47. A sled is pulled along a level path through snow by a rope. A
30-lb force acting at an angle of 40° above the horizontal
moves the sled 80 ft. Find the work done by the force.
48. A boat sails south with the help of a wind blowing in the direction S36"E with magnitude 400 lb. Find the work done by the
wind as the boat moves 120 ft.
~
28. Find two unit vectors that make an angle of 60° with
v = (3,4).
Use a scalar projection to show that the distance from a point
P1(Xh Yl) to the line ax + by + e = 0 is
laxl + bYl + el
.;;;r+b2
29-33 Find the direction cosines and direction angles of the
vector. (Give the direction angles correct to the nearest degree.)
30. (1,-2,-1)
29. (3,4,5)
31. 2i
+ 3j
33. (c,e,e),
32. 2i - j
- 6k
Ifr= (x,y,z),a=
(aha2,a3},andb=
(b1,b2,b3),show
that the vector equation (r - a) • (r - b) = 0 represents a
sphere, and find its center and radius.
~
Find the angle between a diagonal of a cube and one of its
edges.
wheree>O
the scalar and vector projections of b onto a.
3S. a = (3, -4),
b = (5,0)
36. a = (1,2), b = (-4, I)
37. a = (3,6, -2),
38. a=
so.
+ 2k
34. If a vector has direction angles a = 7T/4 and f3 = 7T/3, find the
third direction angle y.
j@Find
Use this formula to find the distance from the point (- 2, 3) to
the line 3x - 4}' + 5 = O.
(-2,3,-6),
b = (1,2, 3)
b=
(5,-1,4)
52. Find the angle between a diagonal of a cube and a diagonal of
one of its faces.
53. A molecule of methane, CH4, is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed
by the H-C-H
combination; it is the angle between the
lines that join the carbon atom to two of the hydrogen atoms.
Show that the bond angle is about 109.5". [Hint: Take the
vertices of the tetrahedron to be the points (1,0,0), (0, 1, 0),
J gFind
the cross product a X b and verify that it is orthogonal
to both a and b.
l.a=(6,0,-2),
b=(0,8,0)
2.a=(I,I,-I),
b=(2,4,6)
3. a = i
+ 3j
4. a = j
+ 7k,
- 2k,
I u X v I and determine whether u X v is directed into
ili'epGe or out of the page.
15.
~vl=8
b = -i + 5k
b = 2i - j
.~~o
"<,
+ 4k
5. a = i - j - k, b = ~i + j
6. a = it e'j
.r('i'4:i5)pind
+ ~k
n6:1 The figure shows a vector a in the xy-plane and a vector b in
the direction of k. Their lengths are I a I = 3 and I b I = 2.
(a) Find [a X bl.
(b) Use the right-hand rule to decide whether the components
of a X b are positive, negative, or 0.
+ e-1k, b = 2i + e'j - e-1k
'/ra\
~ If a = i - 2k and b = j + k, find a X b. Sketch a, b, and
a
X
b as vectors starting at the origin.
9-12 Find the vector, not with determinants, but by using proper-
b
ties of cross products.
9. (i X j) X k
10. k X (i - 2j)
II. (j - k) X (k - i)
12. (i
+ j)
X
x
(i - j)
17. Ifa=(1,2,1)
[ITJ State whether each expression is meaningful. If not, explain
why. If so, state whether it is a vector or a scalar.
(a) a' (b X c)
(c) a X (b X c)
(e) (a- b) X (c- d)
(b) a X (b • c)
(d) (a· b) X c
(f) (a X b) . (c X d)
and b=(0,1,3),findaXbandbXa.
18. If a = (3,1,2),
that a
X
b = (-I, 1,0), and c = (0,0, -4), show
(b X c) of. (a X b) X c.
[J!l Find two unit vectors orthogonal to both (I, - I, I) and
(0,4,4).
3. Find a vector
?it orthogonal to 7/, W
(a)
7/=(1,2,3),
(b)
7/
E R3 where:
W=(1,0,3),
= (5,3,7), W = (1,0,0).
4. Solve the following system in terms of
Xl
{
8XI
Xl
vi:
o
X2:
°
+ 4X2 + X3 + X4 =
+ 1x2 + 8X3 + 3X4 = 0
Is it possible to write the solutions using
5. Solve the following system in terms of
and
Xl
Is it possible to write the solutions using
X2
and
and
X2
X4
instead?
X4
instead?
X2:
and
Find the inverse of A:
A
A~
\!)The
of a 3 x 3 matrix A =
determinant
=(
0
-3 ~
)
0 ~~2)
(1)
• is given by det(A)
( adg ~b f~.)
A~
0 ~n
= aei + bf g + cdh -
ceg-
afh - bdi.
8. Let 7/ x W be the cross product of two vectors in space. Prove (7/ x w) x 7/ = 7/2w - (7/. w)7/
(Hint: use that 7/ x W = 7/ x (w + m 7/)) and notice (7/ x w) x 7/ is perpendicular to 7/).
9. A vector -;; ~
f(X)
= ( ~
-b
I'1(i1 Fin d anum
11
~
G)
~a)(~).
°
in space induces a vector function
~c
a
b ers
I(X) ~ -;; x X.
If
X ~
G)'
z
X
sue h t h at t h e matnx.
(12 8-
X
8 _4)x
b e singu
.
larv
i not inverti
.
ibl e.
ar, I.e.
then
11. Let P
= (Xl, YI) and Q = (X2' Y2) two g(iV; P~intS1)inthe plane
the line determined by P and
Q is det( Xl YI
x2
y+z
12. Show that
x+z
X
1 ) = O.
1
=0
z
1
13. Let M be an (a + b x a + b) -block matrix, i.e. M
and 0 is the zero b x a matrix. Prove det(M)
=( ~I~ )
with A a x a, B b x b, C a x b
= det(A)det(B).
14. Calculate the characteristic polynomial fo the following matrices;
COO)
000
COl)
000
000
000
G
0
1
0
~)G
2
G
the equation of
x+Y
Y
1
1
Y2
~.z. Prove that
6
2
~5)
-10
-3
~)G 00)
(~D G 1 n
(!
o
1
7777)
0
o
1
e -1)
9
-1
1
0
0
o
0
1 0
0
5
0
G ~1)
0
1
GU
~
Determinants
and inverse matrices.
Contents:
• Determinants
and inverse matrices.
Recommended exercises: Leling 11, Leling 12.
EXERCISES
~
(j) Compute
the determinant
@
A=
A= (;
C
3 6
2
350
A~ (
~3
) (fD
~
~ ~
~1 )
-1
(l)
0
0
0
1
2. Solve the following systems using determinants,
(a) { a +,6
0
-1
1
1
0
0
3
3
7 3
11 3
3
0
0
-1
1
1
=1
( b) { m - 3n
1)
=0
n=m-1
x+y+z=l
x+y-z=O
x=l
x+y+z=l
(d)
x
+ 2y + z = 1
{ x +y + 3z = 0
x+y+z+w=l
x
:1:
+y
- z - 3w = 0
1
+ 5w =
x-y=O
(l
~1 )
-1
6
4
i.e. ~ule:
a-,6=O
{
A=
~)
-2
9
(1
A~
-1 -1 0
0
0 -1
1
1
0
A=
-1
U
(e)
@
2)
@)
~1 )
A=(;
.../
3
U
/
6
0
1 0
0 0
1 0
1 0
(m)
C
-2
~
1
A= (
®JA=
01)
(@/C
('i)
t
@
~)
A=
i"-
J.:
..;;::
J@
~
of the following matrices:
-~u)
-1
-1
0
6
5
n