Question

Which of the following functions, (a) f(x,y)=(y,x) or (b) f(x,y)=(−y,x), is an isometry that preserves distances between points in E²?

A) (a) f(x,y)=(y,x)
B) (b) f(x,y)=(−y,x)
C) Both (a) and (b) are isometries.
D) Neither (a) nor (b) are isometries.

Answer 1

Both function (a) f(x,y)=(y,x) and function (b) f(x,y)=(−y,x) are isometries that preserve distances between points in E².

Step-by-step explanation:

To determine which function is an isometry that preserves distances between points in E², we need to check if the distance between any two points remains the same after applying the function. Let's consider two points, (x1, y1) and (x2, y2), in E² and calculate their distances before and after applying the functions:

(a) f(x,y)=(y,x):

Distance before: sqrt((x2 - x1)² + (y1 - y2)²)

Distance after: sqrt((y2 - y1)² + (x1 - x2)²)

(b) f(x,y)=(−y,x):

Distance before: sqrt((x2 - x1)² + (y1 - y2)²)

Distance after: sqrt(((-y2) - (-y1))² + (x1 - x2)²) = sqrt((y1 - y2)² + (x1 - x2)²)

Since both functions preserve distances between points, the correct answer is

C) Both (a) and (b) are isometries.