Question

Point A is at `(4 ,1 )` and point B is at `(-9 ,-7 )`. Point A is rotated `(3pi)/2` clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Answer 1

(-1 ,4) and ≈ 1.664

Explanation:

Under a rotation of `(3pi)/2" clockwise about the origin"`

A point (x ,y) → (-y ,x)

Hence point A(4 ,1) → A'(-1 ,4)

We now require to calculate the distance between A and B and between A' and B.
We can do this using the `color(blue)" distance formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 points"`

For A to B let`(x_1,y_1)=(4 ,1)" and " (x_2,y_2)=(-9 ,-7)`

`d=sqrt((-9-4)^2+(-7-1)^2)=sqrt(169+64)≈15.264`

For A' to B`(x_1,y_1)=(-1 ,4)" and " (x_2,y_2)=(-9 ,-7)`

`d=sqrt((-9+1)^2+(-7-4)^2)=sqrt(64+121)≈13.60`

The difference in distance = 15.264 - 13.60 = 1.664