Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Chapter 12: Vectors and the Geometry
of Space
Math 213
Cross Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross
product of u and v is
u ⇥ v = hu2 v3
u3 v2 , u3 v1
u1 v3 , u1 v2
u2 v1 i
Cross Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross
product of u and v is
u ⇥ v = hu2 v3
u3 v2 , u3 v1
u1 v3 , u1 v2
u2 v1 i
Important Notes:
The cross product of two vectors is a vector.
The cross product is only defined for vectors in R3 .
Cross Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i. Then the cross
product of u and v is
u ⇥ v = hu2 v3
u3 v2 , u3 v1
u1 v3 , u1 v2
u2 v1 i
Important Notes:
The cross product of two vectors is a vector.
The cross product is only defined for vectors in R3 .
We remember this formula by writing
u⇥v=
i
j
k
u1 u2 u3
v1 v2 v3
Even though this doesn’t really make sense, it’s a tool to
help us remember.
Example
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = h2, 3, 1i, v = h3, 1, 4i.
Compute u ⇥ v, v ⇥ u, and u ⇥ u.
Example
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = h2, 3, 1i, v = h3, 1, 4i.
Compute u ⇥ v, v ⇥ u, and u ⇥ u.
h2, 3, 1i ⇥ h3, 1, 4i =
i
2
3
= h 12
j k
3 1
1 4
1, (8
= h 13, 5, 11i
3), 2
( 9)i
Example
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u = h2, 3, 1i, v = h3, 1, 4i.
Compute u ⇥ v, v ⇥ u, and u ⇥ u.
h2, 3, 1i ⇥ h3, 1, 4i =
i
2
3
j k
3 1
1 4
= h 12
1, (8
3), 2
( 9)i
= h 13, 5, 11i
h3, 1, 4i ⇥ h2, 3, 1i =
i
3
2
= h1
j k
1 4
3 1
( 12), (3
= h13, 5, 11i
8), 9
2i
Example Solution (continued)
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
So we see that u ⇥ v 6= v ⇥ u.
In fact, in our example (we will see later this holds in
general), u ⇥ v = (v ⇥ u).
Example Solution (continued)
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
So we see that u ⇥ v 6= v ⇥ u.
In fact, in our example (we will see later this holds in
general), u ⇥ v = (v ⇥ u).
h2, 3, 1i ⇥ h2, 3, 1i =
i
2
2
=h 3
j k
3 1
3 1
( 3), (2
= h0, 0, 0i
2), 6
( 6)i
Example Solution (continued)
The cross product of two vectors
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
So we see that u ⇥ v 6= v ⇥ u.
In fact, in our example (we will see later this holds in
general), u ⇥ v = (v ⇥ u).
h2, 3, 1i ⇥ h2, 3, 1i =
i
2
2
=h 3
j k
3 1
3 1
( 3), (2
2), 6
( 6)i
= h0, 0, 0i
It is also true in general that u ⇥ u = 0 for any vector u 2 R3 .
Orthogonality
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Theorem
The vector u ⇥ v is orthogonal to both u and v.
Orthogonality
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Theorem
The vector u ⇥ v is orthogonal to both u and v.
Proof.
To see u ⇥ v is orthogonal to u, we compute the dot product of
u ⇥ v and u:
u ⇥ v · u = (u2 v3
= u1 u2 v3
u3 v2 )u1 + (u3 v1
u1 u3 v2 + u2 u3 v1
u1 v3 )u2 + (u1 v2
u1 u2 v3 + u1 u3 v2
u2 v1 )u3
u2 u3 v1
=0
A similar calculation shows that u ⇥ v is also orthogonal to v.
Right-hand Rule
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
If a and b are positioned such that their tails have the same
initial point, then the direction that the cross product points
is given by a right-hand rule.
If you curl the fingers of your right hand from a to b (through
an angle less than 180 ), then your thumb points in the
direction of a ⇥ b.
Magnitude of Cross Product Vector
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Theorem
If ✓ is the angle between a and b (so 0  ✓  ⇡), then
|a ⇥ b| = |a||b| sin ✓
(Proof is in text but an elaborated version is found in “Area of
parallelogram proof” on Canvas).
Magnitude of Cross Product Vector
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Theorem
If ✓ is the angle between a and b (so 0  ✓  ⇡), then
|a ⇥ b| = |a||b| sin ✓
(Proof is in text but an elaborated version is found in “Area of
parallelogram proof” on Canvas).
The area of the parallelogram above is A = |a||b| sin ✓.
Therefore, the length of the cross product vector a ⇥ b is equal to
the area of the parallelogram determined by a and b.
Application of Previous Theorem
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Another application of this theorem is to use it to find the
distance between a line ` and a point P not on the line.
We pick two points Q and R on the line and find the vectors
!
!
QR and QP.
Application of Previous Theorem
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Another application of this theorem is to use it to find the
distance between a line ` and a point P not on the line.
We pick two points Q and R on the line and find the vectors
!
!
QR and QP.
The distance D between P and ` is
!
D = |QP| sin ✓
!
!
where ✓ is the angle between QP and QR. But
!
!
! !
|QP ⇥ QR| = |QP||QR| sin ✓
Therefore
!
!
|QP ⇥ QR|
D=
!
|QR|
Properties of the Cross Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u, v, w 2 R3 , and let k 2 R. Then
1
2
3
4
5
6
u⇥v=
v⇥u
(k u) ⇥ v = k (u ⇥ v) = u ⇥ (k v)
u ⇥ (v + w) = u ⇥ v + u ⇥ w
(u + v) ⇥ w = u ⇥ w + v ⇥ w
u · (v ⇥ w) = (u ⇥ v) · w
u ⇥ (v ⇥ w) = (u · w)v
(distributive from the left)
(distributive from the right)
(scalar triple product)
(u · v)w
Properties of the Cross Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
Let u, v, w 2 R3 , and let k 2 R. Then
1
2
3
4
5
6
u⇥v=
v⇥u
(k u) ⇥ v = k (u ⇥ v) = u ⇥ (k v)
u ⇥ (v + w) = u ⇥ v + u ⇥ w
(u + v) ⇥ w = u ⇥ w + v ⇥ w
u · (v ⇥ w) = (u ⇥ v) · w
u ⇥ (v ⇥ w) = (u · w)v
(distributive from the left)
(distributive from the right)
(scalar triple product)
(u · v)w
The proofs of all of these properties follow from the definitions of
the cross product and dot product.
Scalar Triple Product
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
We can write the scalar triple product, a · (b ⇥ c) as the
following determinant:
a · (b ⇥ c) =
a1 a2 a3
b1 b2 b3
c1 c2 c3
Scalar Triple Product
Chapter 12:
Vectors and
the Geometry
of Space
Consider the parallelepiped determined by the vectors a, b,
and c as pictured below.
Math 213
Section 4:
The Cross
Product
The area of the base parallelogram is A = |b ⇥ c|.
Scalar Triple Product
Chapter 12:
Vectors and
the Geometry
of Space
Consider the parallelepiped determined by the vectors a, b,
and c as pictured below.
Math 213
Section 4:
The Cross
Product
The area of the base parallelogram is A = |b ⇥ c|.
The height h of the parallelepiped is |a||cos ✓| where ✓ is the
angle between b ⇥ c and a. (We have to use |cos ✓| in case
✓ > ⇡2 .)
Scalar Triple Product
Chapter 12:
Vectors and
the Geometry
of Space
Consider the parallelepiped determined by the vectors a, b,
and c as pictured below.
Math 213
Section 4:
The Cross
Product
The area of the base parallelogram is A = |b ⇥ c|.
The height h of the parallelepiped is |a||cos ✓| where ✓ is the
angle between b ⇥ c and a. (We have to use |cos ✓| in case
✓ > ⇡2 .)
So the volume of the parallelepiped is
V = Ah = |b ⇥ c||a||cos ✓| = |a · (b ⇥ c)|
Homework
Section 12.4
Chapter 12:
Vectors and
the Geometry
of Space
Math 213
Section 4:
The Cross
Product
1-37 eoo, 53