12.4 The Cross Product

Report
Chapter 12 – Vectors and the
Geometry of Space
12.4 The Cross Product
12.4 The Cross Product
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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
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Cross Product as Determinants

To make Definition one easier, we will
use the notation of determinants. A
determinant of order 2 is defined by
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Cross Product as Determinants

A determinant of order 3 is defined in
terms of second order determinates
as shown below.
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Cross Product as Determinants
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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
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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.
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Visualization

The Cross Product
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Theorems
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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)
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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)
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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:
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Definition – Volume of a
Parallelepiped
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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)
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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
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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.

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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
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Demonstrations
Feel free to explore these
demonstrations below.
 Cross Product of Vectors
12.4 The Cross Product
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