Question

Find the matrix of the rotation in \(\mathbb{R}^{3}\) about the \(x\) axis through the angle \(\theta\) (from the positive \(y\) axis to the positive \(z\) axis).

Short answer

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

Step-by-step solution

Understand Rotation in 3D

Rotating a point or object in 3D space involves using a specific matrix to transform the coordinates. We are tasked with finding the rotation matrix that performs a rotation about the x-axis.

Identify Rotation Axis

The rotation is about the x-axis. In 3D, this transformation will leave the x-coordinate unchanged and only alters the y and z-coordinates.

Formulate the Rotation Matrix

For a rotation about the x-axis through an angle \( \theta \), the standard rotation matrix is given by: \[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \ 0 & \cos(\theta) & -\sin(\theta) \ 0 & \sin(\theta) & \cos(\theta) \end{bmatrix} \] This matrix rotates points in the yz-plane while leaving the x-component of the vector as is.

Verify Your Matrix

Ensure that the matrix derived satisfies the properties of a rotation matrix. This matrix has an orthonormal set of vectors as rows and should have a determinant of 1, confirming that it is a proper rotation matrix.

Key concepts

3D rotation

Rotating objects in three-dimensional space is a common requirement in fields like computer graphics, robotics, and aerospace. A 3D rotation changes the position of a point or an object within 3D space while keeping its structure intact.

Key factors of a 3D rotation include:

• The axis about which the object is being rotated.
• The angle of rotation.
• The direction of rotation (clockwise or counterclockwise).

To describe this transformation mathematically, we use rotation matrices, which are an essential tool for linear transformations in 3D space. Each matrix corresponds to a rotation around a specific axis (x, y, or z) through a defined angle. The use of matrices makes it possible to efficiently compute the new coordinates of points after rotation, ensuring a smooth transformation of the entire object.

Understanding 3D rotations is fundamental for accurately modeling and representing three-dimensional objects from different angles.

x-axis rotation

Rotating around the x-axis is one of the simplest forms of rotation in 3D space because it involves altering only the y and z coordinates while the x coordinate remains unchanged.

This type of rotation can be visualized as spinning an object around an imaginary line that runs through the x-axis. The formula for the rotation matrix around the x-axis is:

\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \ 0 & \cos(\theta) & -\sin(\theta) \ 0 & \sin(\theta) & \cos(\theta) \end{bmatrix} \]
This matrix ensures that the object is rotated correctly while maintaining stability in its x-dimension.

• The first row \([1, 0, 0]\) signifies that x-values remain constant.
• The second and third rows involve the sine and cosine of the angle \(\theta\), affecting the rotation in the yz-plane.

Using this matrix allows us to rotate points or objects efficiently around the x-axis, a critical necessity in many 3D applications.

orthonormal matrices

Orthonormal matrices are a special category of matrices that excel in rotation operations because of their robust mathematical properties. They have rows and columns that are both orthogonal and normalized, meaning they have a magnitude of one.

In the context of rotation matrices, this orthonormality ensures:

• The transformation preserves angles and distances between points, keeping the object’s shape intact.
• The rows and columns of a rotation matrix are mutually perpendicular, a key requirement for 3D transformations.

An orthonormal matrix's inverse is equivalent to its transpose, simplifying computations in scenarios where undoing transformations is necessary. This attribute is fundamental when performing multiple sequential rotations, as it allows operations to be reversed easily.

Thus, working with orthonormal matrices assures accuracy and consistency in computational geometry tasks.

determinant of rotation matrix

The determinant of a rotation matrix is a crucial concept that assures the legitimacy of the transformational matrix as a true rotation matrix. For any proper rotation matrix, the determinant should always equal 1.

This property is significant because:

• A determinant of 1 indicates that the volume and orientation of the object are preserved.
• A value of -1 would suggest that the matrix improperly mirrors or distorts the object.

Verifying that the determinant is 1 is a simple yet essential check to confirm the accuracy of your rotation matrix, especially in complex 3D modeling tasks.

Hence, ensuring the determinant of the rotation matrix is 1 helps maintain the integrity and authenticity of rotational transformations in three-dimensional spaces.