Question

Triangle ABC is rotated to create the image A'B'C'.


Which rule describes the transformation?

Answer 1

Answer on Question #58344 – Math – Analytic Geometry

Question

Triangle ABC is rotated to create the image A'B'C'. Which rule describes the transformation?

Solution

In two dimensions (in a plane)

For example, in a plane the triangle $ABC$ is rotated about $O$ through $\varphi$ angle in an counterclockwise direction.

Triangle rotated about point $O$

Usually rotations on a coordinate grid are considered to be counterclockwise, unless otherwise stated.

In analytic geometry a counterclockwise rotation in Cartesian coordinates is expressed by the following formulae:

$$x ^ {\prime} = x \cos \varphi - y \sin \varphi$$ $$y ^ {\prime} = x \sin \varphi + y \cos \varphi$$

If point A has coordinates $(x_A, y_A)$, then after a counterclockwise rotation through $\varphi$ angle the coordinates of point A' will be

$$x _ {A} ^ {\prime} = x _ {A} \cos \varphi - y _ {A} \sin \varphi$$ $$y _ {A} ^ {\prime} = x _ {A} \sin \varphi + y _ {A} \cos \varphi$$

Particular cases are as follows:

- Rotation of $0{}^{\circ}$ on coordinate axes

$$x _ {A} ^ {\prime} = x _ {A}$$ $$y _ {A} ^ {\prime} = y _ {A}$$

- Rotation of $90{}^{\circ}$ on coordinate axes

$$x _ {A} ^ {\prime} = - y _ {A}$$ $$y _ {A} ^ {\prime} = x _ {A}$$

- Rotation of $180{}^{\circ}$ on coordinate axes (in either direction it is a half-turn)

$$x _ {A} ^ {\prime} = - x _ {A}$$ $$y _ {A} ^ {\prime} = - y _ {A}$$

- Rotation of $270{}^{\circ}$ on coordinate axes ($270{}^{\circ}$ counterclockwise rotation is the same as a $90{}^{\circ}$ clockwise direction)

$$x _ {A} ^ {\prime} = y _ {A}$$ $$y _ {A} ^ {\prime} = - x _ {A}$$

In analytic geometry a clockwise rotation in Cartesian coordinates is expressed by the following formulae:

$$x ^ {\prime} = x \cos \varphi + y \sin \varphi$$ $$y ^ {\prime} = - x \sin \varphi + y \cos \varphi$$

In three dimensions (in a space)

If point A has coordinates $(x_A, y_A, z_A)$, then after rotation trough $\varphi$ angle about the x axis the coordinates of point A' will be

$$x _ {A} ^ {\prime} = x _ {A}$$ $$y _ {A} ^ {\prime} = y _ {A} \cos \varphi - z _ {A} \sin \varphi$$ $$z _ {A} ^ {\prime} = y _ {A} \sin \varphi + z _ {A} \cos \varphi$$

If point A has coordinates $(x_A, y_A, z_A)$, then after rotation trough $\varphi$ angle about the y axis the coordinates of point A' will be

$$x _ {A} ^ {\prime} = x _ {A} \cos \varphi + z _ {A} \sin \varphi$$ $$y _ {A} ^ {\prime} = y _ {A}$$ $$y _ {A} ^ {\prime} = - x _ {A} \sin \varphi + z _ {A} \cos \varphi$$

If point A has coordinates $(x_A, y_A, z_A)$, then after rotation trough $\varphi$ angle about the z axis the coordinates of point A' will be

$$x _ {A} ^ {\prime} = x _ {A} \cos \varphi - y _ {A} \sin \varphi$$ $$y _ {A} ^ {\prime} = x _ {A} \sin \varphi + y _ {A} \cos \varphi$$ $$z _ {A} ^ {\prime} = z _ {A}$$

Any other rotation in three-dimensional space can be obtained from these three.

Answer:

Rule which describes the transformation in a plane (where $\varphi$ is an angle of rotation):

$$x ^ {\prime} = x \cos \varphi - y \sin \varphi$$ $$y ^ {\prime} = x \sin \varphi + y \cos \varphi$$

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