Cross Products in Action

An Example Involving 3x3 Determinants
Question:
Compute the cross product,
u
×
v
, of the following two vectors:
u
= < 3, 1, 0 >,
v
= < 5, 2,
−
1>
Solution:
But computing the value of what is called the ”determinant of a matrix”, can be tricky. We'll leave
the theory for another course, but
for a 3x3 matrix, there there are two common methods:
_________________________________________
Method #1:
Using the Diagonals
Copy the first two columns to the right of the matrix.
Compute the product along the diagonal, down to the right, and add.
Compute the products along the diagonal, down and to the left, and subtract.
_________________________________________
Page #
1 of
3
3, 1,
0
5, 2, 1
3
1
0
5
2
1
i
j
k
uv
3
1
0
3
1
(1)( 1)
(0)(5)
(3)2
(1)5
(0)(2)
(3)( 1)
5
2
1
5
2
( 1 0)
(0 3)
(6 5)
< 1, 3, 1
ijk
i
j
ij
k
k
i
j
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View Full DocumentMethod #2:
Using Minors
Write each of the 2x2 minors corresponding to
i
,
j
and
k
.
Compute the value of each minor.
(call them M
11
, M
12
, M
13
respectively.)
Compute
M
11
i
−
M
12
j
+ M
13
k
But what's a minor?
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 Winter '10
 BEN
 Determinant, Vectors, Method

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