This section is divided into a number of subsections, links to which are:
Parameterizing rotations and the orientation of frames is one of the important parts in 3D geometry. Describing and managing rotations in 3D space is a somewhat more diﬃcult task, compared with the relative simplicity of rotations in the plane. We will explore two methods for dealing with rotation in two following subsections, Euler angles and quaternions.
There is a very important class of transformations known as orthogonal. They are generated by orthogonal matrices (or orthonormal matrices):
3D Rotations
Suppose that you are a pilot, such that the xaxis points to your left, the yaxis points ahead of you, and the zaxis points up. This is the coordinate frame. Then a rotation about the xaxis, denoted by φ, is called the pitch. A rotation about the yaxis, denoted by θ, is called roll. A rotation about the zaxis, denoted by ψ, is called yaw. Euler’s theorem states that any position in space can be expressed by composing three such rotations, for an appropriate choice of (φ, θ, ψ (see next section).
In 3D, rotation occurs about an axis rather than a point, with the term axis taking on its more commonplace meaning of a line about which something rotates. An axis of rotation does not necessarily have to be one of the three standard axes.
When Rotation occurs about a fixed point other than the origin, use the following three step approach:
 Move fixed point to origin,
 Rotate,
 Move fixed point back.

The vector v_{∥}
is the portion of v that is parallel to n:
\[ {\bf v}_{\} = \left( {\bf v} \bullet \hat{\bf n} \right) \hat{\bf n} . \]
 The vector u_{⊥} is the portion of v that is perpendicular to n: \( \displaystyle {\bf v} = {\bf v}_{\} + {\bf v}_{\perp} . \)

The vector w is mutually perpendicular to v_{∥} and v_{⊥}
and has the same length as v_{⊥}. So
\[ {\bf w} = \hat{\bf n} \times {\bf v}_{\perp} . \]

Verify directly that the following 3 × 3 matrices define rotation transformation along the line through the origin and the point (1, 1, 1).
 A rotation by π/6 (= 30°) is \( \displaystyle \quad \frac{1}{3} \begin{bmatrix} \left( 1 + \sqrt{3} \right) & 1 & \left( 1  \sqrt{3} \right) \\ \left( 1  \sqrt{3} \right) & \left( 1 + \sqrt{3} \right) & 1 \\ 1 & \left( 1  \sqrt{3} \right) & \left( 1 + \sqrt{3} \right) \end{bmatrix} ; \)
 A rotation by π/3 (= 60°) is \( \displaystyle \quad \frac{1}{3} \begin{bmatrix} 2&2&1 \\ 1&2&2 \\ 2&1&2 \end{bmatrix} ; \)
 A rotation by π/2 (= 90°) is \( \displaystyle \quad \frac{1}{3} \begin{bmatrix} 1& \left( 1 + \sqrt{3} \right) & \left( 1  \sqrt{3} \right) \\ \left( 1  \sqrt{3} \right) & 1 & \left( 1 + \sqrt{3} \right) \\ \left( 1 + \sqrt{3} \right) & \left( 1  \sqrt{3} \right) & 1 \end{bmatrix} ; \)
 A rotation by 2π/3 (= 120°) is \( \displaystyle \quad \begin{bmatrix} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{bmatrix} . \)
 For a given angle θ, you can write six matrices that define rotations along coordinate vectors i, −i, j, −j, k, and −k. Their inverse matrices give six more matrix expressions. Another twelve expressions are formed by putting −θ for θ. Break these twenty four matrix expressions into a separate lists so that the expressions in each list are necessarily equal but any two matrices from different lists are not equal in general. For example, R(i, θ) and (R(−j, θ))^{−1} are equal and so are in the same list.
 Find all 3 × 3 rotation matrices that are also diagonal.
 Show that a nonzero 3 × 3 matrix P is a rotation if and only if the rows of P have the same crossproduct relation as i, j, k; for example, 1row × 2row = 3roe.
 Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
 Dunn, F. and Parberry, I. (2002). 3D math primer for graphics and game development. Plano, Tex.: Wordware Pub.
 Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), Computer Graphics: Principles and Practice (2nd ed.), Reading: AddisonWesley, ISBN 0201121107
 Matrices and Linear Transformations
 Rogers, D.F., Adams, J. A., Mathematical Elements for Computer Graphics, McGrawHill Science/Engineering/Math, 1989.
 Watt, A., 3D Computer Graphics, AddisonWesley; 3rd edition, 1999.