Question

The vector `vec A` is on a coordinate plane. The plane is then rotated counterclockwise by `phi`. How do I find the components of `vec A` in terms of the components of `vec A` once the plane is rotated?

Original coordinate plane:

`vecA = veci A_x + vecj A_y`

Rotated plane:

`vecA = veci' A_x' + vecj' A_y'`

So,

`veci ' = veci cos phi + vecj sin phi`
`vecj ' = -veci sin phi + vecj cos phi`

`vec A = veci' A_x' + vecj' A_y = vec i(A_x' cos phi - A_y' sin phi) + vec j (A_x' sin phi + A_y' cos phi)`

(I skipped a few steps above.)

`vecA = veci A_x + vecj A_y`, so

`A_x = A_x' cos phi - A_y' sin phi`

`A_y = A_x' sin phi + A_y' cos phi`

However, apparently `A_y = -A_x' sin phi + A_y' cos phi`. I'm not sure where I went wrong?

Answer 1

see below

Explanation:

The matrix `R(alpha)` will rotate CCW any point in the xy-plane through an angle `alpha` about the origin:

• `R(alpha)=((cos alpha , -sin alpha ), (sin alpha , cos alpha))`

But instead of rotating CCW the plane, rotate CW the vector `mathbf A` to see that in the original x-y coordinate system, its co-ordinates are:

`mathbf A' = R(-alpha) mathbf A`

`implies mathbf A = R( alpha) mathbf A'`

`implies ((A_x),(A_y)) = ((cos alpha , -sin alpha ), (sin alpha , cos alpha)) ((A'_x),(A'_y))`

IOW, I think your reasoning looks good.