Chapter 14, Problem 12CR

(a)

To determine

To find: The image of $\left(3,1\right)$ under the transformation $R_x\circ R_y$ .

Answer to Problem 12CR

The image of $\left(3,1\right)$ is $\left(-3,-1\right)$ .

Explanation of Solution

Given information: The point is $\left(3,1\right)$ and transformation is $R_x\circ R_y$ .

Calculation:

If the line of reflection is x -axis then the reflection is as follows:

  $R_x:\left(x_1,y_1\right)\to \left(x_1,-y_1\right)$

If the line of reflection is y -axis then the reflection is as follows:

  $R_y:\left(x_1,y_1\right)\to \left(-x_1,y_1\right)$

The transformation of the point if the line of reflection is y -axis, then

  $R_y:\left(3,1\right)\to \left(-3,1\right)$

Now, the image of $R_y:\left(3,1\right)\to \left(-3,1\right)$ if the line of reflection is x -axis is:

  $R_x\circ R_y:\left(3,1\right)\to \left(-3,-1\right)$

Therefore, the coordinates of the image point is $\left(-3,-1\right)$ .

(b)

To determine

To find: The image of $\left(3,1\right)$ under the transformation $R_y\circ H_O$ .

Answer to Problem 12CR

The image of $\left(3,1\right)$ is $\left(3,-1\right)$ .

Explanation of Solution

Given information: The point is $\left(3,1\right)$ and transformation is $R_y\circ H_O$ .

Calculation:

If the line of reflection is y -axis then the reflection is as follows:

  $R_y:\left(x_1,y_1\right)\to \left(-x_1,y_1\right)$

The image of a point for half-turn about the origin is:

  $H_O:\left(x_1,y_1\right)\to \left(-x_1,-y_1\right)$

The image of given point for half-turn about the origin is:

  $H_O:\left(3,1\right)\to \left(-3,-1\right)$

Now, the image of $\left(-3,-1\right)$ if the line of reflection is y -axis is:

  $R_y\circ H_O:\left(-3,-1\right)\to \left(3,-1\right)$

Therefore, the coordinates of the image point is $\left(3,-1\right)$ .

(c)

To determine

To find: The image of $\left(3,1\right)$ under the transformation $R_x\circ {\Re }_{O,-90}$ .

Answer to Problem 12CR

The image of $\left(3,1\right)$ is $\left(1,3\right)$ .

Explanation of Solution

Given information: The point is $\left(3,1\right)$ and transformation is $R_x\circ {\Re }_{O,-90}$ .

Calculation:

The rotation of a point about the center $O$ is:

  ${\Re }_{O,-90}:\left(3,1\right)\to \left(1,-3\right)$

If the line of reflection is x -axis then the reflection is as follows:

  $R_x:\left(x_1,y_1\right)\to \left(x_1,-y_1\right)$

The image of point $\left(1,-3\right)$ if the line of reflection is x -axis is:

  $R_x\circ {\Re }_{O,-90}:\left(3,1\right)\to \left(1,3\right)$

Therefore, the coordinates of the image point is $\left(1,3\right)$ .