Rotation

Understanding Rotation

Rotation is a geometric transformation that turns every point of an object around a specific center point by a certain angle. This transformation preserves the congruence (shape and size) of the object, but its orientation can change.

Key aspects of rotation:

• Center of Rotation (C): The fixed point around which the rotation occurs.
• Angle of Rotation ($\theta$): The amount of turn. If the angle is positive, the rotation is counter-clockwise. If the angle is negative, the rotation is clockwise.

Definition of Rotation

Given a center point $C$ and a directed angle $\theta$. Rotation with center $C$ by an angle $\theta$, denoted by $\rho_{C,\theta}$ or $R(C,\theta)$, is defined as a transformation that maps:

1. Point $C$ to itself ($C'=C$).
2. Any point $P$ to a point $P'$ such that $CP = CP'$ (the distance from the center to the point is equal to the distance from the center to the image) and the angle formed by ray $\vec{CP}$ and ray $\vec{CP'}$ is $\theta$.

Rotation about the Origin

A common special case is rotation about the origin $O(0,0)$.

If a point $P(x,y)$ is rotated about the origin $O(0,0)$ by an angle $\theta$, its image coordinates $P'(x',y')$ can be calculated using the following formulas:

Rotating a Point by 90°

A point $B(0,4)$ is rotated about the origin $(0,0)$ by $90^\circ$. Determine its image.

Here, $x=0$, $y=4$, and $\theta = 90^\circ$.

We know $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$.

Using the formulas:

Thus, the image of point $B(0,4)$ is $B'(-4,0)$.

Rotating a Line by 90°

Determine the image of the line $y=2x$ rotated about the origin $(0,0)$ by $90^\circ$.

Let $P(x,y)$ be any point on the line $y=2x$. Its image, $P'(x',y')$, after a $90^\circ$ rotation about the origin is:

From this, we get $y = -x'$ and $x = y'$.

Substitute $x = y'$ and $y = -x'$ into the original line equation $y=2x$:

Replacing $x'$ and $y'$ back to $x$ and $y$, the equation of the image line is $-x = 2y$ or $y = -\frac{1}{2}x$.

Exercises

1. A point $A(3,0)$ is rotated about the origin $(0,0)$ by $90^\circ$. Determine its image.
2. Determine the image of the line $y=4x$ rotated about the origin $(0,0)$ by $90^\circ$.
3. Point $P(-2, -5)$ is rotated about the origin $O(0,0)$ by $180^\circ$. Determine the coordinates of its image!

Key Answers

1. Point $A(3,0)$, $\theta = 90^\circ$. $x=3, y=0$.

   $$x' = 3 \cos 90^\circ - 0 \sin 90^\circ = 3(0) - 0(1) = 0$$

   $$y' = 3 \sin 90^\circ + 0 \cos 90^\circ = 3(1) + 0(0) = 3$$

   Thus, its image is $A'(0,3)$.

2. Line $y=4x$, $\theta = 90^\circ$.

   $$x' = -y$$

   $$y' = x$$

   So $y = -x'$ and $x = y'$.

   Substitute into the line equation: $(-x') = 4(y')$.

   Image line equation: $-x = 4y$ or $y = -\frac{1}{4}x$.

3. Point $P(-2,-5)$, $\theta = 180^\circ$. $x=-2, y=-5$.

   $$\cos 180^\circ = -1$$

   $$\sin 180^\circ = 0$$
   $$x' = (-2) \cos 180^\circ - (-5) \sin 180^\circ = (-2)(-1) - (-5)(0) = 2 - 0 = 2$$

   $$y' = (-2) \sin 180^\circ + (-5) \cos 180^\circ = (-2)(0) + (-5)(-1) = 0 + 5 = 5$$

   Thus, the image of point P is $P'(2,5)$.