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VECTORS, TENSORS, AND MATRICES
I Main Topics
A Vector length and direction
B Vector Products
C Tensor notation vs. matrix notation
II Vector Products
A Vector length: A = A x
+ A
y
+ A
z
B A vector A can be defined by its length |A| and the direction of a unit
vector a that is in the same direction as A. The unit vector a has x,y,z
components A x i A , A y j A , and A z k A , respectively, where i,j, and k are
unit vectors along the x,y, and z axes, respectively.
A = A a.
C Example: If A = 0i + 3j + 4k , then A = 0
= 25 = 5 , and
a =
i +
j +
k.
II Products of Vectors
A Dot product: A • B = M
1 A and B are vectors, and M is a scalar corresponding to a length.
2 If unit vectors a and b parallel vectors A and B, respectively, and the
angle from a to b (and from A to B) is θ ab
, then, recalling that
cos θ ab
= cos −θ ab
( ) =^ cos^ θ
ba
( ) =^ cos( )^ θ^ …
a a • b = cos θ = b • a
b A • B = A a • B b = A B a ( • b ) = A B (cos θ)
c Example: If A = 2 i + 0 j + 0 K , and B = 0i + 2j + 0K, A • B = ( ) 2 ( ) 2 cos 90
o ( ) =^0
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3 If b is a unit vector, then A • b (or b • A ) is the length of the projection
of A onto the direction defined by b.
4 Dot product tables of Cartesian basis vectors
i j k
B
x
i
B
y
j
B
z
k
i • 1 0 0
A
x
i •
A
x
B
x
j • (^0 1 )
A
y
j • 0
A
y
B
y^0
k • 0 0 1
A
z
k • 0 0
A
z
B
z
5 A • B = A
x
i + A y
j + A z
k
B
x
i + B y
j + B z
k
= A
x
B
x
+ A
y
B
y
+ A
z
B
z
6 For unit vectors e r
and e s
along axes of a Cartesian frame
a e r
= 1 if r = s
b e r
= 0 if r ≠ s
7 In Matlab, C = A • B is performed as C=A(:)’B(:)or C=sum(A.B)
8 Uses in geology for dot products: all kinds of projections
B Cross product: A × B = C
1 C is a vector perpendicular to both A and B, so C is perpendicular to
the plane containing A and B. C points in the direction of our thumb if
the other fingers on your right hand first point in the direction of A
and then curl to point in the direction of B. (i.e., A, B, and C form a
right-handed set). As a result, A × B = − B × A.
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6 A × B =
i j k
A
x
A
y
A
z
B
x
B
y
B
z
7 For unit vectors e r
and e s
along axes of a Cartesian frame
a e p
× e q
= e r
if p , q = 1,2 or 2,3 or 3,
b e r
× e q
= − e p
if r , q = 3,2 or 2,1 or 1,
b e p
× e q
= 0 if p = q
8 In Matlab, C = A × B is performed as C=cross(A,B)
9 Uses in geology for cross products: finding poles to planes in three-
point problems; finding fold axes from poles to bedding.
C Scalar triple product: ( A,B, C ) = A • ( B × C ) = V
1 The vector triple product is a scalar (i.e., a number) that corresponds
to a volume.
2 |V| is the volume of a parallelepiped with edges along A, B, and C.
(BxC) gives the area of the base, and the dot product of this with A
gives the base times the component of A normal to the base (i.e., the
base times the height). The absolute value of V guarantees that the
volume is non-negative.
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3 V = A • ( B × C ) = A •
i j k
B
x
B
y
B
z
C
x
C
y
C
z
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
A
x
B
y
C
z
− B
z
C
y
A
y
B
x
C
z
− B
z
C
x
A
z
B
x
C
y
− B
y
C
x
4 The determinant of a 3x3 matrix gives the volume of a parallelepiped.
5 In Matlab, V = A • ( B × C ) is performed as V = sum(A.*cross(B,C))
6 Use in geology: solutions of equations, estimating volume of ore
bodies
7 If at least two of the vectors A,B,C are parallel to each other, then
A,B,C cannot define a parallelepiped, at least two rows of the matrix
in (3) are linearly dependent, and the determinant of (3) is zero, and
the three planes defined by A,B,C will not intersect in a unique point
8 Proof that C = A × B is perpendicular to the plane of A and B
a If C is not perpendicular to the AB plane, then C must be non-
perpendicular to both A and C, i.e., A • C ≠ 0 and A • B ≠ 0.
b A • C = A • ( A × B ) = B • ( A × A ) = 0
c B • C = B • ( A × B ) = A • ( B × B ) = 0
d The postulate that C is not perpendicular to the AB plane thus is
disproved, so C is perpendicular to the AB plane.
D Invariants
1 Quantities that do not depend on the orientation of a coordinate
system.
2 Examples
a Dot product of two vectors (a length)
b Scalar triple product (a volume)
