Question

Point A is at `(3 ,-2 )` and point B is at `(5 ,-4 )`. 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

(2 , 3) , ≈ 4.79

Explanation:

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

a point (x , y) → (-y , x )

hence A (3 , -2) → A' (2 , 3 )

To find the change in distance , requires to calculate the length of AB and A'B , and subtract them to find the change.

We can calculate the lengths 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 coordinate points "`

For length AB let `(x_1,y_1)=(3,-2)" and " (x_2,y_2)=(5,-4)`

`d_(AB) = sqrt((5-3)^2 + (-4+2)^2)=sqrt(4+4) ≈ 2.83`

For A'B let `(x_1,y_1)=(2,3)" and " (x_2,y_2)=(5,-4)`

`d_(A'B) = sqrt((5-2)^2 + (-4-3)^2)=sqrt(9+49) ≈ 7.62`

hence , change in length = 7.62 - 2.83 = 4.79