5. Rotation tensor

Now, we will learn about rotation tensors. They are tensors that are related to physical rotation of objects. They can be used to rotate vectors as well as tensors. It should have the property such that after rotation, vectors and tensors do not change their magnitude but only direction.

This chunk introduces rotation tensors as mathematical tools used to represent physical rotations. They maintain the size (magnitude) of vectors and tensors they rotate but alter their direction. This is essential in mechanics, where understanding how objects change orientation without changing size is crucial for calculations.

Think of a spinning top: as it spins, its height (magnitude) remains constant, but its orientation (direction) changes. The rotation tensor captures this idea mathematically.

Let us consider two sets of orthonormal triads. One can always transform a set of triads into another through a unique rotation or what we call a unique rotation tensor.

This chunk emphasizes the concept of orthonormal triads, which consists of three vectors that are perpendicular to one another and have unit length. Rotation tensors facilitate transforming one orthonormal triad into another, underscoring their role in maintaining orthogonality during rotation.

Imagine holding a three-dimensional object with three axes marked. When you rotate the object, those axes (triads) change orientation but remain perpendicular as you turn it. This transformation can be described using a rotation tensor.

Mathematically, we can write $\hat{R} = R e, \forall i = 1,2,3.$ The matrix form of this rotation tensor turns out to be an orthonormal matrix.

Here, the mathematical notation shows how to express the relationship between rotated vectors and their original form using a rotation tensor (R). The tensor is an orthonormal matrix, meaning it preserves lengths and angles during the transformation.

Consider rotating a Rubik's Cube. Each face of the cube maintains its square shape (orthonormal structure) while repositioning relative to the other faces during the rotation. This is analogous to how the rotation tensor mathematically maintains structure while changing orientation.

Orthonormal tensors and their matrix forms have the following properties: (a) R Rᵀ = Rᵀ R = I (an identity tensor), (b) det(R) = 1.

These properties define how rotation tensors behave mathematically. The first property indicates that applying the rotation tensor and its transpose results in an identity matrix, meaning there are no changes post-rotation. The second property ensures the determinant remains 1, indicating the volume and shape remain unchanged.

If you were to spin a basketball perfectly in place without altering its size or distortion, it’s akin to maintaining an identity state through rotation—a concept reinforced by the properties of the orthonormal tensors.

Let us consider a specific example and see how an actual rotation matrix looks like.

This chunk introduces an example to illustrate how a rotation matrix operates in a specific coordinate system. It explains how to determine the matrix form for a given rotation based on specific angles.

If you imagine rotating the camera angle while filming a scene, the rotation matrix would mathematically describe how the camera's angle shifts while keeping the frame's proportions constant.

We can find individual coefficients in terms of θ using dot product definition.

This part explains how to define individual components of the rotation matrix based on angles using the dot product. It emphasizes the calculation aspect to derive exact values for the rotation based on known angles.

Imagine setting the angle on a protractor. Each degree on the protractor helps you determine the precise angle for rotating something in space, akin to deriving coefficients for the rotation matrix.

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