Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
Cross-Products and
Volumes
1
...
, how can we find a third vector w that is ortho
gonal to both u and v? This problem arises naturally in many ways
...
In physics, it is observed that the force on an
electrically charged particle moving in a magnetic field is in the direction orthogonal
to the velocity of the particle and to the vector describing the magnetic field
...
If w is orthogonal to both u and v, it must satisfy the equations
U �V=Ut W1 + U2W2 + U3W3=0
·
V
·
W=V1W1 + V2W2 + V3W3=0
In Chapter 2, we shall develop systematic methods for solving such equations for w1, w2, w3
...
·
=
[
U2V3 - U3V2
U3V1 - U1V3
u1v2 - u2v1
]
Also notice from the form of the equations that any multiple of w would also be orthogonal
to both i1 and v
...
Unlike the dot product of two vectors, which is a scalar, the cross-product of
two vectors is itself a new vector
...
The cross-product is a construction that is defined only in IR
...
)
The formula for the cross-product is a little awkward to remember, but there are
many tricks for remembering it
...
Since the formula can be
difficult to remember, we recommend checking the answer by verifying
that it is orthogonal to both i1 and v
...
Let us see how this works for simple cases
...
3
...
These simple examples also suggest some of the general properties of the cross
product
...
3
and
t E JR
...
One rule we might expect does not in fact hold
...
xx Cfx Z>
*
exxy)x t
This means that the parentheses cannot be omitted in a cross-product
...
See
Problem F3 in Further Problems at the end of this chapter
...
W hat is the length of the
cross-product of a and v? We give the answer in the following theorem
...
3
and e be the angle between it and
V, then llilx vii = llitll llVll sine
...
We have
vll2 = (U V3 - U3V )2 + (U3V1 - U1V3)2 + (U1V - U V1)2
2
2
2
2
Expand by the binomial theorem and then add and subtract the term Cuivi+u�v�+u�v�)
...
v)2= lli1112llvll2 - lli1112llvll2 cos2e
= lli1112llvll2(1 - cos2 B) = llz1112lli1112 sin2e
and the result follows
...
5
...
Assuming that i1 x v f:
...
Take the length of z1 to be the base of
the parallelogram; the altitude is the length of perpa v
...
5
...
and V =
[il
·
EXAMPLE3
Find the area of the paraIIelogram determined by a =
[-�]
and ii =
[3l
and
[-: l
Solution: By Theorem 2, the area is
EXERCISE4
Find the area of the parallelogram determined by"=
=
ii
LH
Some Problems on Lines, Planes, and Distances
The cross-product allows us to answer many questions about lines, planes, and dis
tances in JR3
...
2, the vector equation of a
plane was given in the form 1
jJ + sil + t\!, where {il, v} is linearly independent
...
Therefore, it will be
given by it= i1 xv
...
Find a scalar equation of the plane that contains
these lines
...
_,
P, Q, and R lie in the plane, then so do the directed line segments PQ
PR
...
are parallel, their intersection will be a line
...
It can therefore be obtained
as the cross-product of the two normals
...
EXAMPLE6
Find a vector equation of the line of intersection of the two planes x1
and 2x1 - X2
+
3x3
=
+
x2 - 2x3
=
3
6
...
Hence, a
6
...
+
x3
=
-2
The Scalar Triple Product and Volumes zn �3 The three vectors
a, v, and w in JR
...
5
...
Is there an expression for the volume of the parallelepiped in
terms of the three vectors? To obtain such a formula, observe that the parallelogram
a and v can be regarded as the base of the solid of the parallelepiped
...
With respect to this base, the altitude of the solid is the
length of the amount of w in the direction of the normal vector ft
a xv to the base
...
=
II proJ,1 w
...
lw
...
(it xv)I
11a xvii
Thus, to get the volume of the parallelepiped, multiply this altitude by the area of the
base to get
...
ca xv)I
x11a xvii
Ila xvii
called the scalar triple product of
=
lw ca xv)I
·
w, it, and v
...
axv
altitude
base area
Figure 1
...
19
11 proj,1 wll
11axVil
The parallelepiped with adjacent edges
lw
...
a, v, w
has volume given by
The sign of the scalar triple product also has an interpretation
...
as the
Some other
(in that order) is right-handed, if and only if
the scalar triple product is positive
...
It is often useful to note that
w
...
cv xw) = v
...
EXAMPLE 7
Find the volume of the pamllelepiped determined by the vectors
Solution: The volume Vis
V=
m l[:H=rn [�JHJ
=
=2
[H [H [ n
and
=
PROBLEMS 1
...
+iJ {l]
{�] +�J
{�H�l
[�H=ii
(+�I {!l
[!Hll
nl Ul [=H
c
c
c
A2 ld =
v =
and
w =
Check
by calculation that the following general properties
hold
...
cv xw) = w
...
cv xw) -v
...
( c)
(•) ,
=
tains each set of points
...
- X3 = 5 and
2x1 - 5x2 +-
...
(•)
(b)
(c)
(d)
(e)
[�Hrrnl
u1rn m
nrnrn1
[_H urn1
n1nrn1
AS What does it mean, geometrically, if il·(vxw) = O?
A9 Show that (il - v) x(il+v) = 2(il xv)
...
Hl {�]
m +�l
]
[ �� ] {i
![ ]· Ul
(a)
(c )
( el
V =
+!H�l
Hl
�
c
2
B2 Ut U =
B4 Determine the scalar equation of the plane with
and W =
Ch �k
by calculation that the following general properties
hold
...
(b) i1xv = -vx i1
(c) i1 x 2w = 2(i1x w)
(a) P(S,2,1), Q(-3,2,4),R(8,1,6)
Cd) ax cv+ w) = axv+ ax w
(b) P(S, -1,2), Q(-1,3, 4),R(3,1,1)
Ce) i1
...
caxv)
Cf) a
...
cax w)
(c) P(0,2,1), Q(3,-1,1),R(l,3,0)
(d) P(l,5,-3), Q(2,6,-1),R(l,O, 1)
B3 Calculate the area of the parallelogram determined
by each pair of vectors
...
3
...
[-;]
m [-�l
and
B6 Determine a vector equation of the line of intersec
-
tion of the given planes
...
(a)
(b)
(c)
(d)
HlUlUI
liHH []
lH lH Ul
lH lH Ul
Conceptual Problems
Dl Show that if X is a point on the line through P and
Q, then x x (q -
p
OP, and q = OQ
...
�
=
p x if,
where x =
OX,
D2 Consider the following statement: "If i1 f
...
" If the statement is true,
prove it
...
D3 Explain why u x (v x w) must be a vector that sat
isfies the vector equation x
=
sv + tW
...
3 such that
(a) ax (v x w)
=
(b) ax (v x w) *
(u xv) x w
(u xv) x w
CHAPTER REVIEW
Suggestions for Student Review
Organizing your own review is an important step to
by giving examples
...
It is much more valu
the concept of linear independence
...
2)
able than memorizing someone else's list of key ideas
...
You should also be able to
give (or, even better, create) instructive examples
...
3
...
(Section 1
...
tive checklist; instead, they suggest the kinds of activ
Explain the connection between the formal defini
ities and questioning that will help you gain a confident
tion of linear dependence and an intuitive geometric
grasp of the material
...
Why is linear
1 Find some person or persons to talk with about math
ematics
...
2)
ment is a small price for learning
...
3 and the dot product in JR
...
Use examples to
illustrate
...
3)
get lots of practice in writing answers independently
...
Be sure you do your share of asking
and answering
...
(Section 1
...
(Albert Einstein once said, "If
you can't explain it simply, you don't understand it
well enough
...
1)
v is defined
in terms of the dot product
...
Define the part of a vector x perpendicular to v and
verify (in the general case) that it is perpendicular to
v
...
4)
10 Explain with a picture how projections help us to
solve the minimum distance problem
...
4)
11 Discuss the role of the normal vector to a plane in de
termining the scalar equation of the plane
...
n
...
3 that are lines, planes, and
to a vector equation for the plane and from a vector
all of JR
...
Show that there is only one subspace in
equation of the plane to the scalar equation
...
3 that does not have infinitely many vectors in it
...
3 and 1
...
2)
5 Show that the subspace spanned by three vectors in
JR
...
3,
12 State the important algebraic and geometric proper
ties of the cross-product
...
5)
Chapter Quiz
propriate level of difficulty for a test on this material
...
{[�], [ �]}
E4 Pro,e thatS
-
=
{[::J
ElO Each of the following statements is to be inter
is a basis for JR
...
3
E IR I a,x, + a,x, + a3x3
=
d
3
is a subspace of JR
...
=
is true, and if so, explain briefly
...
(i) Any three distinct points lie in exactly one
plane
...
•
...
(iv) The dot product of a vector with itself cannot
be zero
...
3
...
ES Determine the cosine of the angle between
ii
and
E9 Prove that the volume of the parallelepiped deter
mined by il, v, and w has the same volume as the
parallelepiped determined by (il +kV), v, and w
...
E3 Show that
[�] nl
ES Determine a non-zero vector that is orthogonal to
Note: Your instructor may have different ideas of an ap
(v) For any vectors x and y, proj1 y
proj
...
(vi) For any vectors x and y, the set {proj1 y, perp1 y}
=
is linearly independent
...
Illustrate your
method of calculation with a sketch
...
E7 Find the point on the hyperplane x1 + x + x3 +
2
x4
1 that is closest to the point P(3, -2, 0, 2) and
=
determine the distance from the point to the plane
...
Some explore topics beyond the material discussed in
the text
...
5
...
v)w
...
=
=
...
" Either prove
=
=
=
=
the statement or give a counterexample
...
3
...
,
F4 Prove that
(a) it· v
±11a + vll2 - ±llit - vll2
211a112 + 211v112
Cb) 11a + vli2 + 11a - v112
=
=
(c) Interpret (a) and (b) in terms of a parallelogram
determined by vectors il and v
...
2, two lines fail to have a point of intersection
only if they are parallel
...
3, a pair of
lines can fail to have a point of intersection even if
they are not parallel
...
3 are called
skew
...
Show that two skew
lines do lie in parallel planes