As a domain expert in geometry and transformations, I can provide a detailed explanation on the commutativity of translations and rotations.

In the realm of Euclidean geometry, transformations are operations that change the position or shape of an object. Two common types of transformations are translations and rotations.
Translation is a transformation that moves every point of a shape the same distance in the same direction.
Rotation, on the other hand, is a transformation that turns a shape around a fixed point, known as the center of rotation, by a certain angle.

The concept of commutativity in mathematics refers to the property that the order in which operations are performed does not change the result. For example, addition and multiplication are commutative operations because \(a + b = b + a\) and \(a \times b = b \times a\) for any numbers \(a\) and \(b\).

Now, let's consider the question of whether translations and rotations are commutative. To explore this, we can examine what happens when we apply these transformations to a point in a coordinate system.

Translation can be represented by a vector \(\mathbf{v} = (h, k)\), where \(h\) is the horizontal component and \(k\) is the vertical component of the translation. When we translate a point \(P(x, y)\) by vector \(\mathbf{v}\), the new position \(P'\) is given by:
\[ P'(x', y') = P(x, y) + \mathbf{v} = (x + h, y + k) \]

Rotation about the origin by an angle \(\alpha\) can be represented using rotation matrices. For a point \(P(x, y)\), the new position \(P''(x'', y'')\) after rotation is:
\[ \begin{bmatrix} x'' \\ y'' \end{bmatrix} = \begin{bmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]
\[ P''(x'', y'') = (x\cos(\alpha) - y\sin(\alpha), x\sin(\alpha) + y\cos(\alpha)) \]

If we were to apply a translation followed by a rotation, the sequence of operations would be:
1. Translate the point \(P(x, y)\) to \(P'(x + h, y + k)\).
2. Rotate \(P'\) about the origin by angle \(\alpha\) to get \(P''(x'', y'')\).

The resulting point \(P''\) would be:
\[ P''(x'', y'') = ((x + h)\cos(\alpha) - (y + k)\sin(\alpha), (x + h)\sin(\alpha) + (y + k)\cos(\alpha)) \]

If translations and rotations were commutative, then it would not matter whether we first translated and then rotated, or first rotated and then translated. However, as we can see from the above expressions, the resulting point \(P''\) after first translating and then rotating is not the same as the point we would get if we first rotated and then translated.

To illustrate this with an example, let's say we have a point \(P(1, 1)\), a translation vector \(\mathbf{v} = (2, 3)\), and a rotation angle \(\alpha = 90^\circ\) (or \(\pi/2\) radians). Applying the translation first and then the rotation would yield a different result than applying the rotation first and then the translation.

Therefore, based on the mathematical definitions and the example provided, we can conclude that **translations and rotations are not commutative**. The order in which these transformations are applied does affect the final position of a point in space.

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Let us use a form similar to the homogeneous coordinates. ... However, if translation (h,k) is applied first followed by a rotation of angle a (about the coordinate origin), we will have the following: Therefore, rotation and translation are not commutative!read more >>