Question

Point A is at `(3 ,7 )` and point B is at `(5 ,-4 )`. 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

(7 ,-3) , ≈ 8.944

Explanation:

The first step is to find the new coordinates of point A , which I will name A'.

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

a point (x ,y) → (y ,-x)

hence A (3 ,7) → A' (7 ,-3)

To calculate the change in length of AB with A'B use 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 using A(3 ,7) and B(5 ,-4)

`d_(AB)=sqrt((5-3)^2+(-4-7)^2)=sqrt125≈11.18`

Length of A'B using A'(7 ,-3) and B(5 ,-4)

`d_(A'B)=sqrt((5-7)^2+(-4+3)^2)=sqrt5≈2.236`

change in length = 11.18 - 2.236 = 8.944