Chapter 12 – Vectors and the
Geometry of Space
12.4 The Cross Product
12.4 The Cross Product
1
Definition – Cross Product
Note: The result is a vector. Sometimes the cross
product is called a vector product. This only
works for three dimensional vectors.
12.4 The Cross Product
2
Cross Product as Determinants

To make Definition one easier, we will
use the notation of determinants. A
determinant of order 2 is defined by
12.4 The Cross Product
3
Cross Product as Determinants

A determinant of order 3 is defined in
terms of second order determinates
as shown below.
12.4 The Cross Product
4
Cross Product as Determinants
12.4 The Cross Product
5
Example 1 – pg. 814 #5

Find the cross product a x b and
verify that it is orthogonal to both a
and b.
a  i  jk
1
1
b  i  j k
2
2
12.4 The Cross Product
6
Theorem 5

The direction of axb is given by the right hand rule: If your
fingers of your right hand curl in the direction of a rotation
of an angle less than 180o from a to b, then your thumb
points in the direction of axb.
12.4 The Cross Product
7
Visualization

The Cross Product
12.4 The Cross Product
8
Theorems
12.4 The Cross Product
9
Example 2

For the below problem, find the
following:
◦ a nonzero vector orthogonal to the plane
through the points P, Q, and R.
◦ the area of triangle PQR.
P(2,1,5)
Q(-1,3,4)
R(3,0,6)
12.4 The Cross Product
10
Theorem 8

Note: The cross product is not commutative
ixjjxi

Associative law for multiplication does not hold.
(a x b) x c  a x (b x c)
12.4 The Cross Product
11
Definition – Triple Products

The product a  (b x c) is called the
scalar triple product of vectors a, b,
and c. We can write the scalar triple
product as a determinant:
12.4 The Cross Product
12
Definition – Volume of a
Parallelepiped
12.4 The Cross Product
13
Example 3 – pg. 815 # 36

Find the volume of the parallelepiped
with adjacent edges PQ, PR, and PS.
P(3,0,1)
R(5,1,-1)
Q(-1,2,5)
S(0,4,2)
12.4 The Cross Product
14
Torque



Cross product occurs often in physics.
Let’s consider a force, F, acting on a rigid body at a point
given by a position vector r. (i.e. tightening a bolt by
applying force to a wrench). The torque  is defined as
=rxF
and measures the tendency of the body to rotate about the
origin.
The magnitude of the torque vector is
||= |r x F| = |r||F|sin
12.4 The Cross Product
15
Example 4 – pg. 815 #41

A wrench 30 cm long lies along the
positive y-axis and grips a bolt at the
origin. A force is applied in the
direction <0,3,-4> at the end of the
wrench. Find the magnitude of the
force needed to supply 100 N m of
torque to the bolt.

12.4 The Cross Product
16
More Examples
The video examples below are from
section 12.4 in your textbook. Please
watch them on your own time for
extra instruction. Each video is
about 2 minutes in length.
◦ Example 1
◦ Example 2
◦ Example 5
12.4 The Cross Product
17
Demonstrations
Feel free to explore these
demonstrations below.
 Cross Product of Vectors
12.4 The Cross Product
18