# Composition of rotational transformations

• seyma
In summary, when basic rotations are combined to find composite rotational transformations, if the rotation is about one of the principal axes of OXYZ (the fixed frame), the previous resultant rotation matrix is premultiplied by the new rotational transformation. Similarly, if the rotation is about one of the principal axes of OUVW (the moving frame), the previous resultant rotation matrix is postmultiplied by the new rotational transformation. This can be proven using the orthonormality of the transformation matrix.
seyma
Show that when basic rotations are combined to ﬁnd composite rota-
tional transformations, if the rotation is about one of the principal axes of
OXYZ (the ﬁxed frame) the previous resultant rotation matrix is premulti-
pled by the new rotational transformation, and if the rotation is about one
of the principal axes of OUVW (the moving frame) the previous resultant
rotation matrix is postmultipled by the new rotational transformation.

## Homework Equations

I know that I can write fixed frame coordinates in terms of moving frame that is the transition matrix which is composed of unit vectors of two coordinates. I call this R. Then I
call fixed frame rotation A and moving frame rotation B. then this becomes A = RB. also for any rotation matrix call R' I can write AR' = R'B.

## The Attempt at a Solution

Also I know all transformation matrix is orthonormal. However I cannot know how to prove.

Let's first consider the rotation about one of the principal axes of OXYZ (the fixed frame). Then the previous resultant rotation matrix is premultiplied by the new rotational transformation. This means that A = RB, where R is the transition matrix composed of the unit vectors of two coordinates and B is the new rotational transformation. To prove this we first use the orthonormality of the matrix. Since both A and R are orthonormal, we have A'A = I, where I is the identity matrix. Similarly we have R'R = I. Using these two equations and applying them to the above equation gives us A'AR' = I. Since A' = A and R' = R, we get AA'R = I. Since A'R = B, we get AB = I. Thus we can conclude that the previous resultant rotation matrix is premultiplied by the new rotational transformation. Now let's consider the rotation about one of the principal axes of OUVW (the moving frame). Then the previous resultant rotation matrix is postmultiplied by the new rotational transformation. This means that AR' = R'B, where R is the transition matrix composed of the unit vectors of two coordinates and B is the new rotational transformation. To prove this we first use the orthonormality of the matrix. Since both A and R are orthonormal, we have A'A = I, where I is the identity matrix. Similarly we have R'R = I. Using these two equations and applying them to the above equation gives us A'AR' = I. Since A' = A and R' = R, we get A'AR' = I. Since A'R = B, we get AR' = B. Thus we can conclude that the previous resultant rotation matrix is postmultiplied by the new rotational transformation.

## 1. What is meant by "composition of rotational transformations"?

The composition of rotational transformations refers to the process of applying multiple rotational transformations to an object or function in a specific order. This results in a final transformation that combines the effects of each individual transformation.

## 2. How is the composition of rotational transformations different from a single rotational transformation?

A single rotational transformation involves rotating an object or function around a fixed point, whereas the composition of rotational transformations involves rotating an object or function around multiple fixed points in a specific order.

## 3. What is the mathematical representation of composition of rotational transformations?

The composition of rotational transformations can be represented using matrix multiplication. Each individual transformation is represented by a rotation matrix, and the final transformation is calculated by multiplying these matrices in the specified order.

## 4. What are some common applications of composition of rotational transformations?

Composition of rotational transformations is commonly used in computer graphics, robotics, and animation. It can also be applied in physics and engineering to understand the motion of rotating objects.

## 5. Is the order of rotational transformations important in composition?

Yes, the order of rotational transformations is important in composition. Changing the order of transformations can result in a different final transformation, as rotations do not commute. This means that the order in which the transformations are applied affects the final result.

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