Question

Point A is at `(-1 ,-8 )` and point B is at `(-5 ,3 )`. Point A is rotated `pi/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

(-8 ,1) and ≈ 8.104

Explanation:

Under a rotation of `pi/2` clockwise about O.

a point (x ,y) → (y ,-x)

hence point A(-1 ,-8) → A' (-8 ,1)

To find the change in distance between AB and A'B we require to calculate the length of AB and A'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"`

length of AB`(x_1,y_1)=(-1 ,-8)" and " (x_2,y_2)=(-5,3)`

`d_(AB)=sqrt((-5+1)^2+(3+8)^2)=sqrt(16+121)≈11.705`

length of A'B
`(x_1,y_1)=(-8 ,1)" and " (x_2,y_2)=(-5, 3)`

`d_(A'B)=sqrt((-5+8)^2+(3-1)^2)=sqrt(9+4)≈3.601`

hence the change in length = 11.705 - 3.601 = 8.104