Chapter 14.1, Problem 20WE

(a)

To determine

To plot: The points $P\left(4,5\right)$ , $Q\left(-3,2\right)$ and $R\left(-3,-1\right)$ , and their images.

Explanation of Solution

Given information: The given point are $P\left(4,5\right)$ , $Q\left(-3,2\right)$ and $R\left(-3,-1\right)$ . The mapping is $S:\left(x,y\right)\to \left(x,0\right)$ .

Graph:

Consider the mapping $S:\left(x,y\right)\to \left(x,0\right)$ .

The image of point $P\left(4,5\right)$ is:

  $P:\left(4,5\right)\to P^{\prime }\left(4,0\right)$

The image of point $Q\left(-3,2\right)$ is:

  $Q:\left(-3,2\right)\to Q^{\prime }\left(-3,0\right)$

The image of point $R\left(-3,-1\right)$ is:

  $R:\left(-3,-1\right)\to R^{\prime }\left(-3,0\right)$

  

Figure (1)

Interpretation: From the graph it can be observed that the distance between the point and its image is the x -coordinate of the point.

(b)

To determine

To find: Whether $S$ appear to be an isometry or not.

Answer to Problem 20WE

The mapping $S$ does not appear an isometry.

Explanation of Solution

Given information: The given point are $P\left(4,5\right)$ , $Q\left(-3,2\right)$ and $R\left(-3,-1\right)$ . The mapping is $S:\left(x,y\right)\to \left(x,0\right)$ .

Calculation:

From part (a), the images of the given points are $P^{\prime }\left(4,0\right)$ , $Q^{\prime }\left(-3,0\right)$ and $R^{\prime }\left(-3,0\right)$ .

For isometry the distance between $P,Q,R$ and $P^{\prime },Q^{\prime },R^{\prime }$ must be the same.

The distance between the point P and Q is:

  $\begin{array}{c}PQ=\sqrt{{\left(4+3\right)}^2+{\left(5-2\right)}^2}\\ =\sqrt{7^2+3^2}\\ =\sqrt{49+9}\\ =\sqrt{58}\end{array}$

The distance between the point $P^{\prime }$ and $Q^{\prime }$ is:

  $\begin{array}{c}P\text{'}Q\text{'}=\sqrt{{\left(4+3\right)}^2+{\left(0-0\right)}^2}\\ =\sqrt{7^2+0}\\ =\sqrt{49}\\ =7\end{array}$

Since $PQ\ne P\text{'}Q\text{'}$ . So, mapping $S$ is not an isometry.

Therefore, the mapping $S$ does not appear an isometry.

(c)

To determine

To check: Whether $S$ is a transformation or not.

Answer to Problem 20WE

The mapping $S$ is a transformation.

Explanation of Solution

Given information: The mapping $S:\left(x,y\right)\to \left(x,0\right)$ .

Calculation:

If the image of origin is origin, then the mapping is said to be transformation. The image of origin is:

  $S:\left(0,0\right)\to \left(0,0\right)$

Hence, the mapping $S$ is a transformation.