The document discusses determinants of matrices and their geometric interpretations. It begins by defining a matrix as a rectangular table of numbers or formulas. It then explains that the determinant of a 2x2 matrix gives the signed area of the parallelogram defined by the row vectors of the matrix. The sign of the determinant indicates whether the parallelogram is swept clockwise or counterclockwise. Finally, it generalizes these concepts to 3x3 matrices by writing out the formula for the determinant as a sum of signed areas of parallelograms.

1. Determinant and Cross Product

2. A matrix is a rectangular table with numbers
or formulas as entries.
Determinant and Cross Product

3. A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns.
Determinant and Cross Product

4. A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2
4 –1
x 2 –1
6 –2 y2
11 9 –4
Determinant and Cross Product
5 2 –3
4 –1 0
A2X2 = B2X3 = C3X3 =

5. A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2
4 –1
x 2 –1
6 –2 y2
11 9 –4
Determinant and Cross Product
Matrices of sizes n x n, such as A and C above,
are called square matrices.
5 2 –3
4 –1 0
A2X2 = B2X3 = C3X3 =

6. A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2
4 –1
x 2 –1
6 –2 y2
11 9 –4
Determinant and Cross Product
Matrices of sizes n x n, such as A and C above,
are called square matrices. For each square matrix A,
we extract a number called the determinant of A.
5 2 –3
4 –1 0
A2X2 = B2X3 = C3X3 =

7. A matrix is a rectangular table with numbers
or formulas as entries. The size of a matrix is r x c
where r is the number of rows and c is the number of
columns. Following are examples of matrices:
5 2
4 –1
x 2 –1
6 –2 y2
11 9 –4
Determinant and Cross Product
Matrices of sizes n x n, such as A and C above,
are called square matrices. For each square matrix A,
we extract a number called the determinant of A.
5 2 –3
4 –1 0
A2X2 = B2X3 = C3X3 =
Given a 2x2 matrix A the determinant of A is
a b
c d
det = ad – bc. Hence det(A) = –13.det(A) =

8. a b
c d
det = ad – bc
Determinant and Cross Product
The 2x2 determinant
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),

9. a b
c d
det = ad – bc
Determinant and Cross Product
The 2x2 determinant
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
(a, b)
(c, d)

10. a b
c d
det = ad – bc
Determinant and Cross Product
The 2x2 determinant
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
(a, b)
(c, d)
(a+c, b+d)

11. a b
c d
det = ad – bc
Determinant and Cross Product
The 2x2 determinant
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
(a, b)
(c, d)
(a+c, b+d)
i.e. |det(A)| = the area of

12. Determinant and Cross Product
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown

13. Determinant and Cross Product
(a+c, b+d)
(a, b)
(c, d)
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown

14. Determinant and Cross Product
(a+c, b+d)
(a, b)
(c, d)
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown

15. Determinant and Cross Product
(a+c, b+d)
(a, b)
(c, d)
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown
.
.

16. Determinant and Cross Product
(a+c, b+d)
(a, b)
(c, d)
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown and note that area of
.
.
= – 2(
.
+ + )

17. Determinant and Cross Product
(a+c, b+d)
(a, b)
(c, d)
a b
c d
det = ad – bc
The 2x2 determinant
(a, b)
(c, d)
(a+c, b+d)
gives the signed area of the
parallelogram defined by the
row–vectors (a, b) and (c, d),
i.e. |det(A)| = the area of
To see this, we dissect the
rectangle that boxes–in the
parallelogram into regions as
shown and note that area of
.
.
= – 2(
.
+ + )
= ad – bc (check this.)

18. a b
c d
det = ad – bc
Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
gives the following information concerning directions.
D =

19. a b
c d
det = ad – bc
i. If D > 0, then the parallelogram
is formed by sweeping u = < a, b>
in a counterclockwise motion,
i.e. to the left of u.
Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
gives the following information concerning directions.
D =
(a, b)
(c, d)
u
D = ad – bc > 0

20. a b
c d
det = ad – bc
i. If D > 0, then the parallelogram
is formed by sweeping u = < a, b>
in a counterclockwise motion,
i.e. to the left of u.
Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
gives the following information concerning directions.
D =
(a, b)
(c, d)
(a, b)
(c, d)u
u
D = ad – bc > 0
ii. If D < 0, then the parallelogram
is formed by sweeping u = < a, b>
in a clockwise motion,
i.e. to the right of u.
D = ad – bc < 0

21. a b
c d
det = ad – bc
i. If D > 0, then the parallelogram
is formed by sweeping u = < a, b>
in a counterclockwise motion,
i.e. to the left of u.
Determinant and Cross Product
The sign of the 2x2 determinant D of the matrix
gives the following information concerning directions.
D =
(a, b)
(c, d)
(a, b)
(c, d)u
u
D = ad – bc > 0
ii. If D < 0, then the parallelogram
is formed by sweeping u = < a, b>
in a clockwise motion,
i.e. to the right of u.
D = ad – bc < 0
iii. If D = 0, there is no parallelogram,
It’s deformed to a line.

22. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
so from i = < 1, 0> to j = <0, 1> is a

23. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
0 1
1 0
det = –1,
is a clockwise sweep.
so from i = < 1, 0> to j = <0, 1> is a
so from j = <0, 1> to i = <1, 0> is a

24. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
0 1
1 0
det = –1,
is a clockwise sweep.
Hence we say the matrix
c d
preserves orientation if D > 0,
a b
so from i = < 1, 0> to j = <0, 1> is a
so from j = <0, 1> to i = <1, 0> is a

25. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
0 1
1 0
det = –1,
is a clockwise sweep.
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
a b
so from i = < 1, 0> to j = <0, 1> is a
so from j = <0, 1> to i = <1, 0> is a

26. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
0 1
1 0
det = –1,
is a clockwise sweep.
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
a b
reverses orientation if D < 0,
so from i = < 1, 0> to j = <0, 1> is a
so from j = <0, 1> to i = <1, 0> is a

27. 1 0
0 1
det = +1,
Determinant and Cross Product
Example A.
counterclockwise sweep.
0 1
1 0
det = –1,
is a clockwise sweep.
Hence we say the matrix
c d
preserves orientation if D > 0, i.e. facing in the direction
of <a, b>, the vector <c, d> is to the left, just as in our
R2 if we face the x–axis, the y–axis is to the left,
a b
reverses orientation if D < 0, i.e. facing in the direction
of <a, b>, <c, d> is to the right which is the mirror
image to our choice of R2.
so from i = < 1, 0> to j = <0, 1> is a
so from j = <0, 1> to i = <1, 0> is a

28. Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.

29. a b c
d e f
g h i
det
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.

30. a b c
d e f
g h i
det 
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b
d e f d e
g h i g h

31. a b c
d e f
g h i
det 
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b
d e f d e
g h i g h

32. a b c
d e f
g h i
det 
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b
d e f d e
g h i g h

33. a b c
d e f
g h i
det 
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
a b c a b
d e f d e
g h i g h
= aei

34. a b c
d e f
g h i
det 
= aei + bfg
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ +
a b c a b
d e f d e
g h i g h

35. a b c
d e f
g h i
det 
= aei + bfg + cdh
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + +
a b c a b
d e f d e
g h i g h

36. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + –
a b c a b
d e f d e
g h i g h

37. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
+ + + – – –
a b c a b
d e f d e
g h i g h

38. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
1 2 –1
6 –2 2
3 1 –4
Hence det
1 2 –1 1 2
6 –2 2 6 –2
3 1 –4 3 1

+ + + – – –
a b c a b
d e f d e
g h i g h

39. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
1 2 –1
6 –2 2
3 1 –4
Hence det
1 2 –1 1 2
6 –2 2 6 –2
3 1 –4 3 1
8 12 –6

+ + + – – –
a b c a b
d e f d e
g h i g h

40. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
1 2 –1
6 –2 2
3 1 –4
Hence det
1 2 –1 1 2
6 –2 2 6 –2
3 1 –4 3 1
8 12 –6 6 2 –48

+ + + – – –
a b c a b
d e f d e
g h i g h

41. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
1 2 –1
6 –2 2
3 1 –4
Hence det
1 2 –1 1 2
6 –2 2 6 –2
3 1 –4 3 1
8 12 –6 6 2 –48
= 8 + 12 – 6 – (6 + 2 – 48)
= 54

+ + + – – –
a b c a b
d e f d e
g h i g h

42. a b c
d e f
g h i
det 
= aei + bfg + cdh – (ceg + afh + bdi)
Determinant and Cross Product
Let us generalize 2x2 determinants and the associated
geometry to 3x3 determinants.
1 2 –1
6 –2 2
3 1 –4
Hence det
1 2 –1 1 2
6 –2 2 6 –2
3 1 –4 3 1
8 12 –6 6 2 –48
= 8 + 12 – 6 – (6 + 2 – 48)
= 54

+ + + – – –
a b c a b
d e f d e
g h i g h
Following are some of the important geometric and
algebraic properties concerning determinants.

43. Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.

44. a b c
d e f
g h i
If det = D,
Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.
x
z+
<a, b, c>
<d, e, f>
y
<g, h, i>
then D is the signed volume of
D < 0
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.

45. a b c
d e f
g h i
If det = D,
Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.
x
z+
<a, b, c>
<d, e, f>
y
<g, h, i>
then D is the signed volume of
D < 0
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.

46. a b c
d e f
g h i
If det = D,
Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.
x
z+
<a, b, c>
<d, e, f>
y
<g, h, i>
then D is the signed volume of
i. If D > 0, then the row vectors
<a, b, c> → <d, e, f>→ <g, h, i> form a right handed
system as in the case i → j → k.
D < 0
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.

47. a b c
d e f
g h i
If det = D,
Determinant and Cross Product
The geometric significance of the
3x3 determinants is the same as
the 2x2 determinants.
x
z+
<a, b, c>
<d, e, f>
y
<g, h, i>
then D is the signed volume of
i. If D > 0, then the row vectors
<a, b, c> → <d, e, f>→ <g, h, i> form a right handed
system as in the case i → j → k.
ii. If D < 0, then <a, b, c> → <d, e, f>→ <g, h, i>
form a left handed system as in the case i → j → –k.
D < 0
the parallelepiped, i.e. a squashed box,
formed by the row–vectors of the matrix.

48. Determinant and Cross Product
Here are some important properties of determinants.

49. Determinant and Cross Product
x * *
0 y *
0 0 z
i. det = xyz, hence det
1 0 0
0 1 0
0 0 1
= 1
Here are some important properties of determinants.

50. Determinant and Cross Product
x * *
0 y *
0 0 z
i. det = xyz, hence det
det
1 0 0
0 1 0
0 0 1
= 1
a b c
d e f
g h i
ii. Given that det = D, then
d e f
g h i
=
a b c
g h i
det
a b c
d e f= det
ka kb kc
kd ke kf
kg kh ki
= kD
where k is a constant,
Here are some important properties of determinants.

51. Determinant and Cross Product
x * *
0 y *
0 0 z
i. det = xyz, hence det
det
1 0 0
0 1 0
0 0 1
= 1
a b c
d e f
g h i
ii. Given that det = D, then
d e f
g h i
=
a b c
g h i
det
a b c
d e f= det
ka kb kc
kd ke kf
kg kh ki
= kD
where k is a constant, i.e. stretching an edge of the
box by a factor k, then the volume of the box is
changed by the factor k.
Here are some important properties of determinants.

52. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix.

53. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
if det = D, then
a b c
d e f
g h i
det = –D.
d e f
a b c
g h i

54. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
Geometrically, this says that if we switch the order of
any two vector, we change the orientation of the
system defined by the row vectors.
if det = D, then
a b c
d e f
g h i
det = –D.
d e f
a b c
g h i

55. Determinant and Cross Product
3. The determinant changes sign if we swap two rows
in the matrix. For example,
Geometrically, this says that if we switch the order of
any two vector, we change the orientation of the
system defined by the row vectors.
if det = D, then
a b c
d e f
g h i
det = –D.
d e f
a b c
g h i
1 0 0
0 1 0
0 0 1
detSo = 1 and
1 0 0
0 0 1
0 1 0
det = –1
since we changed from a right handed system to a
left handed system following the row vectors.

56. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.

57. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.
Cross Product

58. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
Cross Product

59. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k
a b c
d e f
detu x v =
Cross Product

60. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k
a b c
d e f
detu x v =
i j k
d e f
a b c
detand v x u =
Cross Product

61. Determinant and Cross Product
In fact the sign of the determinant is the algebraic
definition of the right handed system vs. the left handed
system, i.e. a positive determinant means the row
vectors form a right handed system and a negative
determinant means they form a left handed system.
Given two 3D vectors u = <a, b, c>, v = <d, e, f>,
their cross products are defined below
i j k
a b c
d e f
detu x v =
i j k
d e f
a b c
detand v x u =
Cross Product
The cross product is only defined for two 3D vectors,
the product yield another 3D vector.

62. Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
Determinant and Cross Product

63. i j k
1 2 –1
2 –1 3
det
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
Determinant and Cross Product

64. i j k
1 2 –1
2 –1 3
det
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1

Determinant and Cross Product

65. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j

Determinant and Cross Product

66. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product

67. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.

68. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
v
u

69. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
u x v v
u

70. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
u x v v
u
* The length |u x v| is equal to
the area of the parallelogram
defined by u and v.

71. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
u x v v
u
* The length |u x v| is equal to
the area of the parallelogram
defined by u and v.
u x v v
u

72. i j k
1 2 –1
2 –1 3
det
6i –2j –k
Example B. Find the cross product of
u = <1, 2, –1>, v = <2, –1, 3 >.
u x v =
i j k
1 2 –1
2 –1 3
i j
1 2
2 –1
4k i 3j
= 6i – 2j – k – (4k + i + 3j) or that u x v = 5i – 5j – 5k.

Determinant and Cross Product
* u x v is a orthogonal to both
u and v in the direction determined
by the right-hand rule.
u x v v
u
* The length |u x v| is equal to
the area of the parallelogram
defined by u and v.
u x v v
u
|u x v| = area of the

73. Cross Products of i, j, and k
Determinant and Cross Product

74. Cross Products of i, j, and k
Determinant and Cross Product
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.

75. Cross Products of i, j, and k
i x j = k
j x k = i
k x i = j
(Forward ijk)
Determinant and Cross Product
i
jk
+
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.

76. i
jk
Cross Products of i, j, and k
i x j = k
j x k = i
k x i = j
–
i x k = –j
j x i = –k
k x j = –i
(Forward ijk) (Backward kji)
Determinant and Cross Product
i
jk
+
Since the unit coordinate vectors i, j, and k are
mutually perpendicular, we have the following
relations.

77. Determinant and Cross Product
Algebra of Cross Product

78. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
Determinant and Cross Product
Algebra of Cross Product

79. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
Determinant and Cross Product
Algebra of Cross Product

80. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
Determinant and Cross Product
Algebra of Cross Product

81. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
Determinant and Cross Product
Algebra of Cross Product

82. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
Determinant and Cross Product
Algebra of Cross Product

83. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
6. (Triple Scalar Product) u•(v x w) =
u
v
w
det
Determinant and Cross Product
Algebra of Cross Product

84. Let u, v and w be 3D vectors:
1. u x u = u x 0 = 0
2. (Orthogonal Property)
u • (v x u) = 0, v • (v x u) = 0
3. (Anti-commutative) u x v = – (v x u)
4. If k is a scalar, (ku) x v = u x (kv) = k(u x v)
5. (Distributive Law)
(u + v) x w = u x w + v x w
u x (v + w) = u x v + u x w
6. (Triple Scalar Product) u•(v x w) =
u
v
w
det
Determinant and Cross Product
Algebra of Cross Product
In particular |u•(v x w)| = volume of the parallelepiped
defined by u, v, and w.

85. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w)
Determinant and Cross Product

86. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
Determinant and Cross Product

87. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
Determinant and Cross Product

88. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
Determinant and Cross Product

89. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) =
Determinant and Cross Product

90. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
Determinant and Cross Product

91. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
Determinant and Cross Product

92. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
= 6 *
u
v
w
det
Determinant and Cross Product

93. Example C. Given that u = <2, 0, 0>, v = <0, 3, 0>,
and w = <0, 0, 4>. Find
a. u x (2u – 3w) = u x (2u) + u x (–3w)
= 2(u x u) – 3(u x w)
= 2*0 – 3<0, –8, 0> = <0, 24, 0>
b. u • (2v x 3w) = u • [6(v x w)]
= 6[u • (v x w)]
= 6 * = 6 * 24 = 144
u
v
w
det
Volume of the box
formed by u,v and w
Determinant and Cross Product

94. Determinant and Cross Product