Page 40 - Physical Principles of Sedimentary Basin Analysis

22 Properties of porous media

because we have that n = (cos θ, sin θ) and n = (− sin θ, cos θ). This form of writing
x z
the rotation matrix is easily generalized to 3D, where we have that
⎛ ⎞

n · n x n · n y n · n z
x x x

R = ⎝ n · n x n · n y n · n z ⎠ . (2.57)
y y y

n · n x n · n y n · n z
z
z
z
We see that the entries in the rotation matrices are simply the elements of the primed unit
vectors in the unprimed system. This form of the rotation matrix is useful in numerical
applications, where it might be necessary to compute a rotation matrix for each element in

a finite element grid. The scalar product of two unit vectors n and n j is also the cosine
i

of the angle between the unit vectors, because n · n j = cos(α ij ), where α ij is the angle
i

between n and n j . The rotation matrix is therefore sometimes called the direction cosine
i
matrix.
The2Drotationmatrix(2.53) rotates round the y-axis. Such rotations in 3D around the
x-, y- and the z-axes are
⎛ ⎞ ⎛ ⎞
1 0 0 cos β 0 − sin β
R x (α) = ⎝ 0 cos α sin α ⎠ , R y (β) = ⎝ 0 1 0 ⎠ (2.58)
0 − sin α cos α sin β 0 cos β
and
cos γ − sin γ 0
⎛ ⎞
R z (γ ) = ⎝ sin γ cos γ 0 ⎠ . (2.59)
0 0 1
These rotation matrices can be multiplied together:
R(α,β,γ ) = R x (α) R y (β) R z (γ ) (2.60)
where any possible rotation is specified by the three angles α, β and γ . An important point
is that the product (2.60) depends on the order of the multiplication of the rotation matrices.
Exercise 2.12 Show that the length of a vector remains the same after a rotation of the
coordinate system by an angle θ.

Exercise 2.13 Show the following equalities:
T
R(θ) −1 = R(−θ) = R(θ) . (2.61)
Solution: The first equality is obvious because θ − θ = 0. The second equality follows
from expression (2.57) for the rotation matrix, where the inverse rotation is obtained by
interchanging the primed and unprimed basis vectors.
Exercise 2.14 A coordinate system has as unit vectors

1 T 1 T 1 T

n = √ (1, 1, 1) , n = √ (1, −2, 1) , n = √ (1, 0, −1) . (2.62)
z
y
x
3 6 2