1.4 Cross Product 1.4.1 De…nitions

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CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
1.4
Cross Product
1.4.1
De…nitions
The cross product is the second multiplication operation between vectors we will
study. The goal behind the de…nition of this new operation is that we wanted
its result to be a vector, perpendicular to the two vectors we are taking the
cross product of.
Unlike the dot product, the cross product is only de…ned for 3-D vectors. In
this section, when we use the word vector, we will mean 3-D vector.
De…nition 45 (cross product) The cross product also called vector product of two vectors ~u = hux ; uy ; uz i and ~v = hvx ; vy ; vz i, denoted ~u ~v , is de…ned
to be
0
1 0
1 0
1
ux
vx
u y vz u z vy
@ uy A @ vy A = @ uz vx ux vz A
uz
vz
ux vy uy vx
Thus, the cross product of two 3-D vectors is also a 3-D vector.
This formula is not easy to remember. However, if you know about matrices
and the determinant of a matrix, the cross product can be expressed in term of
them. Let us …rst quickly review what they are.
De…nition 46 We only give the de…nition of the determinant of a 2
3 3 matrix.
a
c
1. The determinant of a 2 2 matrix
b
d
, denoted by
a
c
b
d
2 and a
is de…ned
to be
a
c
b
= ad bc
d
2
3
a1 a2 a3
2. The determinant of a 3 3 matrix 4 b1 b2 b3 5denoted by
c1 c2 c3
is de…ned to be
a1
b1
c1
a2
b2
c2
a3
b3
c3
Example 47 Find
1
7
b2
c2
=
a1
=
a1 (b2 c3
b3
c3
a2
c 2 b3 )
b1
c1
b3
c3
a2 (b1 c3
2
3
1
7
2
3
=
(1) (3)
=
3
=
14
11
(7) (2)
+ a3
b1
c1
a1
b1
c1
a2
b2
c2
a3
b3
c3
b2
c2
c1 b3 ) + a3 (b1 c2
c1 b2 )
1.4. CROSS PRODUCT
1
3
4
Example 48 Find
1
3
4
2
1
7
29
2
1
7
3
1
2
3
1
2
1
7
1
2
2
=
1
=
(1) (2
7)
=
5
4 + 51
=
42
3
4
2 (6
1
2
+3
3
4
4) + 3 (21
1
7
4)
Proposition 49 If ~u = hux ; uy ; uz i and ~v = hvx ; vy ; vz i then
~u
~v =
!
i
ux
vx
!
j
uy
vy
!
k
uz
vz
Which makes it much easier to remember.
Proof.
!
i
ux
vx
!
j
uy
vy
!
k
uz
vz
=
(uy vz
uz vy )~i
=
(uy vz
uz vy ; uz vx
= ~u
(ux vz
uz vx ) ~j + (ux vy
ux vz ; ux vy
uy vx )
~v
Example 50 For !
u = h3; 1; 1i and !
v = h4; 7; 2i, compute ~u
~u
~v
=
=
uy vx ) ~k
~v .
!
~i !
j
k
3 1 1
4 7 2
(2
7)~i
(6
!
4) j + (21
!
4) k
= h 5; 2; 17i
The above tells us how to compute the cross product. However, it does
not tell us what the cross product represents. There is a very nice geometric
interpretation of the interpretation of the cross product.
1.4.2
Properties
Theorem 51 Let ~u and ~v denote two non-zero vectors. Then, the following is
true:
1. ~u
~u = ~0
30
CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
2. ~u
~v is perpendicular to both ~u and ~v .
3. k~u
(0
~v k = k~uk k~v k sin where
is the smallest angle between ~u and ~v
).
4. ~u ~v = k~uk k~v k sin !
n where !
n is the unit vector perpendicular to both
~u and ~v whose direction is determined by the right-hand rule.
Remark 52 The above properties tell us that ~u ~v is the vector perpendicular to
both ~u and ~v which direction is given by the right-hand rule and whose magnitude
is k~uk k~v k sin . This is very important. There are many situations in which
one needs to …nd a vector perpendicular to two known vectors. We illustrate it
with examples.
Example 53 Find a unit vector perpendicular to both ~u = h1; 1; 1i and ~v =
h2; 1; 0i.
First, we …nd a vector perpendicular to these two vectors, then we make it into
a unit vector. Such a vector is
~u
~v
= h1; 1; 1i h2; 1; 0i
~i ~j ~k
=
1 1 1
2 1 0
! ! !
= 2j
k
i
= h 1; 2; 1i
A unit vector in the same direction is
h 1; 2; 1i
h 1; 2; 1i
p
n=
=
kh 1; 1; 2ik
6
1 2
1
p ;p ;p
6
6
6
Example 54 Find a vector perpendicular to the plane containing the three
points P : (1; 1; 2), Q : (2; 1; 1) and R : (2; 1; 0).
As long as the three points are not collinear, we can make two not parallel vec!
!
tors from them, for example P Q and QR. A vector perpendicular to the plane
!
!
will be perpendicular to both vectors. Such a vector is P Q QR.
!
P Q = h1; 0; 1i
and
QR = h0; 0; 1i
Therefore,
!
PQ
~i ~j
1 0
0 0
!
QR =
= ~j
=
h0; 1; 0i
~k
1
1
1.4. CROSS PRODUCT
31
Remark 55 Using the de…nition, it is easy to verify that
~i ~j = ~k
~j ~k = ~i
~k ~i = ~j
and
~j
~k
~k
~i
~i =
~j =
~k =
~i
~j
Remark 56 From property 3 of theorem 51, it follows that two non-zero vectors
are parallel if and only if their cross product is ~0.
The cross product satis…es more properties which we will not prove because
they are very tedious.
Theorem 57 Let ~u, ~v , and w
~ be three vectors and a be a scalar. The following
is true:
1. ~u
~v =
2. (a~u)
3. ~u
~v
~u (this tells us that the cross product is not commutative.
~v = a (~u
(~v + w)
~ = ~u
4. (~u + ~v )
w
~ = ~u
~v ) = ~u
(a~v )
~v + ~u
w
~
w
~ + ~v
w
~
Area of a Parallelogram
Consider a parallelogram whose sides are given by the vectors ~u and ~v as shown
in …gure 1.13. Remembering that the area of a parallelogram is the length of
its base times its height, we see that the area A of this parallelogram is
A =
=
k~uk k~v k sin
k~u
~v k
Example 58 Find the area of the parallelogram shown in …gure 1.14.
If we let A = (0; 0; 0), B = (0; 4; 0), C = (0; 1; 3) and D = (0; 5; 3) then
!
Area = AB
!
AC
First, we compute the cross product.
!
AB
!
AC
= h0; 4; 0i
= h12; 0; 0i
h0; 1; 3i
32
CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
Figure 1.13: Area of the paralelogram is k!
u
!
v k = k!
u k k!
v k sin
Therefore
Area = kh12; 0; 0ik
=
1.4.3
12
Triple Products
De…nition 59 Given three non-zero vectors ~u, ~v , and w,
~ the product ~u (~v
is called the scalar triple product of the vectors ~u, ~v , and w.
~
w)
~
Proposition 60 The volume of the parallelepiped determined by the vectors ~u,
~v , and w
~ as shown in …gure 1.15 is the magnitude of their scalar triple product
j~u (~v w)j.
~
Proof. The volume V of a parallelepiped is given by
V = area of the base times height
Suppose the base of the parallelepiped is determined by ~v and w.
~ Let be the
angle ~u makes with the direction perpendicular to the base. Then the height of
the parallelepiped is jk~uk cos j. The area of the base is k~v wk.
~ Therefore,
V
= jk~v
= j~u (~v
wk
~ k~uk cos j
w)j
~
1.4. CROSS PRODUCT
33
Figure 1.14: Find the area
Figure 1.15: Parallelepiped determined by !
u, !
v and !
w
34
CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
Corollary 61 Three non-zero vectors ~u, ~v , and w
~ are coplanar (on the same
plane) if ~u (~v w)
~ = 0:
Remark 62 If instead of thinking of the parallelepiped as having its base determined by ~v and w,
~ we had thought of it as having its base determined by ~u and
~v , then we would have found that its volume was jw
~ (~u ~v )j. But since we are
talking about the same parallelepiped, the two formulas for the volume must be
the same, so we have:
~u (~v w)
~ =w
~ (~u ~v )
(1.7)
Remark 63 The scalar triple product of three non-zero vectors ~u, ~v , and w
~ can
be computed by calculating the determinant
~u (~v
1.4.4
w)
~ =
ux
vx
wx
uy
vy
wy
uz
vz
wz
(1.8)
Summary
The cross product is a very important quantity in mathematics. It can be used
for:
1. Find a vector perpendicular to two non-zero vectors (often used in computer graphics).
2. Find the area of a parallelogram.
3. Find the volume of a parallelepiped.
4. Determine if two non-zero vectors are parallel.
5. Determine if three non-zero vectors are coplanar.
6. Many applications in physics which we will not discuss here.
1.4.5
Vectors and Maple
To handle vectors using Maple 9.5, one must …rst load the LinearAlgebra
package with the command
with(LinearAlgebra);
Once this package is loaded, the following operations can be performed:
De…ning a vector: This is 2done using
the construct h; ; i. For example,
3
1
to de…ne the vector A to be 4 3 5, use
4
A := h1; 3; 4i ;
1.4. CROSS PRODUCT
35
Adding two vectors: Use the usual addition symbol as in
A+B
Scalar Multiplication: Use the usual multiplication symbol as in
2 A
Subtracting two vectors: Use the usual subtraction symbol as in
A
B
Finding the norm of a vector: The norm we de…ned in this class is
called the 2-norm in more advanced mathematics classes because we take
the square root of the sum of the squares of the coordinates. To do this
with Maple, use
N orm(A; 2);
where A is a vector.
Dot product: Given two vectors A and B, their dot product can be
found using
DotProduct(A,B);
or the shortcut
A:B;
Cross product: Given two vectors A and B, their cross product can be
found using
CrossProduct(A,B);
or the shortcut
A &x B;
There must be spaces between A and & as well as between x and B.
Plotting vectors: To plot vectors, one must …rst load the plots package
with the command
with(plots);
To plot the vector A, one would then use
arrow(A, shape=arrow);
The shape parameter is optional. To plot two or more vectors, one must
list the vectors inside square brackets. The command is:
arrow([A,B],shape=arrow);
To …nd all the parameters of the arrow command, use the help facility of
Maple.
36
CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE
1.4.6
Problems
1. Find the length when de…ned of !
u !
v and !
v !
u for the given !
u and
!
v below.
!
! !
! !
(a) !
u =2i
2j
k and !
v = i
k.
!
!
!
!
! !
!
!
(b) u = 2 i
2 j + 4 k and v = i + j
2k.
!
!
(c) !
u = 2 i and !
v = 3j.
(d) !
u =
!
8i
!
!
! !
4 k and !
v =2 i +2j + k.
!
2j
2. Sketch !
u, !
v and !
u
!
!
!
u for !
u = i and !
v = j.
3. Find the area of the triangle determined by P = (1; 1; 2), Q = (2; 0; 1)
and R = (0; 2; 1). Then, …nd a unit vector perpendicular to the plane
P QR.
4. Find the area of the triangle determined by P = (2; 2; 1), Q = (3; 1; 2)
and R = (3; 1; 1). Then, …nd a unit vector perpendicular to the plane
P QR.
5. Verify that (!
u !
v) !
w = (!
v !
w) !
u = (!
w !
u) !
v and …nd the
volume of the parallelepiped determined by these three vectors for !
u =
(2; 0; 0), !
v = (0; 2; 0) and !
w = (0; 0; 2).
6. Verify that (!
u !
v) !
w = (!
v !
w) !
u = (!
w !
u) !
v and …nd the
volume of the parallelepiped determined by these three vectors for !
u =
(2; 1; 0), !
v = (2; 1; 1) and !
w = (1; 0; 2).
!
!
! !
!
7. Let !
u =5i
1j + k, !
v = j
5 k , and !
w =
Which vectors if any are parallel, perpendicular?
!
!
15 i + 3 j
!
3k.
8. Which of the following are always true and which of the following are not
always true? Give reasons.
p
u !
u.
(a) k!
uk= !
!
!
!
(b) u u = k u k.
(c) !
u
!
(d) u
(e) !
u
!
(f) u
(g) (!
u
!
(h) ( u
! !
0 = 0
! !
u= 0.
!
v =!
v
!
!
u = 0.
!
u.
!
!
( v + w) = !
u
!
v)
!
v)
!
v +!
u
!
v = 0.
!
w =!
u (!
v
!
w.
!
w ).
9. Given nonzero vectors !
u, !
v and !
w , use dot product and cross product
notation to describe the following:
1.4. CROSS PRODUCT
37
(a) The vector projection of !
u onto !
v.
!
!
(b) A vector orthogonal to u and v .
(c) A vector orthogonal to !
u
!
v and !
w.
(d) The volume of the parallelepiped determined by !
u, !
v and !
w.
!
!
!
!
(e) A vector orthogonal to u
v and u
w.
(f) A vector of length k!
u k in the direction of !
v.
10. Let !
u, !
v and !
w be vectors. Decide which expressions below make sense
and which do not. Give reasons.
(a) (!
u !
v)
!
!
(b) u ( v
(c) !
u (!
v
!
w.
!
w ).
!
w ).
(d) !
u (!
v !
w ).
11. Cancellation law. If !
u !
v =!
u
!
!
v = w ? Justify your answer.
!
!
w and !
u =
6 0 , does it follow that
12. Find the area of the parallelogram whose vertices are A = (1; 0), B =
(0; 1), C = ( 1; 0) and D = (0; 1).
13. Find the area of the parallelogram whose vertices are A = ( 1; 2), B =
(2; 0), C = (7; 1) and D = (4; 3).
14. Find the area of the triangle whose vertices are A = (0; 0), B = ( 2; 3)
and C = (3; 1)
15. Find the area of the triangle whose vertices are A = ( 5; 3), B = (1; 2)
and C = (6; 2)