Question

Find the matrix of the rotation about the \(y\) axis through the angle \(\theta\) (from the positive \(x\) axis to the positive \(z\) axis).

Short answer

The matrix is \[ R_{y}(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{bmatrix} \].

Step-by-step solution

Understand the Rotation Matrix Concept

In a three-dimensional space, a rotation about the coordinate axes can be represented using a rotation matrix. For rotation about the y-axis, the general form of the rotation matrix is based on the cosine and sine of the angle of rotation, \(\theta\).

Set Up the Rotation Matrix

The rotation matrix for rotating about the y-axis through an angle \(\theta\) is:\[R_{y}(\theta) = \begin{bmatrix}\cos \theta & 0 & \sin \theta \0 & 1 & 0 \-\sin \theta & 0 & \cos \theta\end{bmatrix}\]This matrix will rotate a point counterclockwise when viewed from the positive direction of the y-axis.

Key concepts

3D Transformations

In the realm of mathematics and computer graphics, 3D transformations are crucial in manipulating objects within a three-dimensional space. These transformations allow us to change the position, orientation, or size of objects. They are comprised mainly of translation, which moves an object; scaling, which resizes an object; and rotation, which changes the direction that an object is facing.
A transformation in 3D space can be represented using matrices, which provide a powerful and compact way to apply these changes. Essential to 3D transformations are rotation matrices which are used specifically to change the orientation of an object without affecting its shape or position. Every point in the object is rotated about an axis of rotation.
In our current exercise, we focus on rotation matrices, which are a subset of 3D transformations, to understand how objects can be rotated around one of the coordinate axes - in this case, the y-axis.

Rotation About Axes

Rotating an object means turning it around an axis, which is an imaginary line around which the object spins. In 3D space, you can perform a rotation about any of the three main axes: x-axis, y-axis, or z-axis. These rotations are foundational for tasks such as aligning objects correctly, animating shapes, or simulating real-world physics scenarios.
The key element in performing a rotation is the rotation matrix. For instance, the rotation about the y-axis involves the matrix \[R_{y}(\theta) = \begin{bmatrix}\cos \theta & 0 & \sin \theta \0 & 1 & 0 \-\sin \theta & 0 & \cos \theta\end{bmatrix}\] This matrix transforms all the points in the object's coordinate system by rotating it around the y-axis by an angle \( \theta \). For a positive angle \( \theta \), the rotation is counterclockwise as seen from the positive y-axis, which changes the coordinates of points relative to this axis.
Understanding this concept aids in grasping complex spatial transformations, enhancing one's ability to model or simulate 3D environments.

Coordinate Systems

Coordinate systems are the framework we use to locate points in space. In 3D space, we often use the Cartesian coordinate system, which features three axes intersecting at right angles, typically denoted as x, y, and z. This system helps define every point in space with a set of three coordinates \((x, y, z)\).
In the context of 3D transformations, understanding coordinate systems is vital because rotations and other transformations are usually defined in reference to these axes. Each axis serves as a pivot around which objects can be rotated, and movement along these axes is what defines translation.
When you perform a rotation, the rotation matrix uses these axes to reorient the object's points. For example, as seen with the y-axis rotation matrix, knowing the coordinate system allows us to predict exactly how the object will move. This makes coordinate systems indispensable for both theoretical understanding and practical application in fields like computer graphics and engineering.