MAT 1236
Calculus III
Section 12.4
The Cross Product
http://myhome.spu.edu/lauw
HW…
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WebAssign 12.4
Read 12.5 (Seriously!): The first not too
easy section in Calculus
Preview
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Define a new operation on vectors: The
Cross Product
Unlike the dot product, the cross product
of two vectors is a vector.
Properties of the cross product.
The Right Hand Rule
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FBI
We are Interested in …
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Given 2 vectors, they “span” a plane
Find a vector perpendicular to this plane
The Cross Product
If a  a , a , a and b  b , b , b , the cross product
of a and b is the vector
1
2
3
1
2
3
a  b  a1 , a2 , a3  b1 , b2 , b3
 a2b3  a3b2 , a3b1  a1b3 , a1b2  a2b1
  a2b3  a3b2  i   a3b1  a1b3  j   a1b2  a2b1  k
The Cross Product
The formula is traditionally memorized by
using (formal) determinant expansions
a  b  a1 , a2 , a3  b1 , b2 , b3
 a2b3  a3b2 , a3b1  a1b3 , a1b2  a2b1
  a2b3  a3b2  i   a3b1  a1b3  j   a1b2  a2b1  k
2x2 Determinant Expansions
a b
c d
 ad  bc
3x3 Determinant Expansions
a
b
c
d
g
e
h
f a
h
i
e
f
i
b
d
f
g
i
c
d
e
g
h
3x3 Determinant Expansions
a
b
c
d
g
e
h
f a
h
i
e
f
i
b
d
f
g
i
c
d
e
g
h
3x3 Determinant Expansions
a
b
c
d
g
e
h
f a
h
i
e
f
i
b
d
f
g
i
c
d
e
g
h
3x3 Determinant Expansions
a
b
c
d
g
e
h
f a
h
i
e
f
i
b
d
f
g
i
c
d
e
g
h
The Cross Product
The formula is traditionally memorized by
using (formal) determinant expansions
i
j
k
a  b  a1
b1
a2
b2
a3
b3
a  b  a1 , a2 , a3  b1 , b2 , b3
  a2b3  a3b2  i   a3b1  a1b3  j   a1b2  a2b1  k
Example 1
a  2i  3 j  k , b  i  k
i
ab 
j
k

Expectations
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You are expected to use the above
standard procedure to find the cross
product.
You are expected to show all the steps.
Keep in mind, good practices are key to
minimize the chance of making
mistakes.
Property A
aa  0
i
j
k
a  a  a1
a2
a3 
a1
a2
a3
Property B
a  b is orthogonal to both a and b
 a  b   a 

 a  b   b 
Property B
a  b is orthogonal to both a and b
 a  b   a 

 a  b   b 
In addition, the cross
product obeys the
Right Hand Rule.
Property B (Why?)
a  b is orthogonal to both a and b
i
j
a  b  a1
b1
a2
b2
k
a2
a3 
b2
b3
a3
a1
i
b3
b1
a3
a1
j
b3
b1
a2
b2
k
Example 1 (Verify Property B)
a  2i  3 j  k , b  i  k , a  b  3i  3 j  3k
 a  b   a 


 a b b 


Property C
a  b  a b sin  , 0    
Property C (Why?)
a  b  a b sin  , 0    
In Particular
In Particular
i  j is in the same direction of k and
i j 
Property D
Two nonzero vectors and are parallel if
and only if a  b  0
Property D (Why?)
Two nonzero vectors and are parallel if
and only if a  b  0
ab  0 
Property E
The length of the cross product axb is
equal to the area of the parallelogram
determined by a and b.
A  a b sin   a  b
Example 2
Find a vector perpendicular to the plane
that passes through the points
P(6,0,0) , Q(1,1,1), R(0,0,2)
Example 3
Find the area of the triangle with vertices
P(6,0,0) , Q(1,1,1), R(0,0,2)
Other Properties
Right Hand Rule
Default
Reference only