- #1

domabo

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## Homework Statement

With respect to a given Cartesian coordinate system S , a vector A has components Ax= 5 , Ay= −3 , Az = 0 . Consider a second coordinate system S′ such that the (x′, y′) x y z coordinate axes in S′ are rotated by an angle θ = 60 degrees with respect to the (x, y) coordinate axes in S , (Figure 3.26). (a) What are the components Ax and Ay of vector A in coordinate system S′ ? (b) Calculate the magnitude of the vector using the ( Ax , A y ) components and using the ( Ax' , Ay' ) components. Does your result agree with what you expect?

https://imgur.com/gallery/b06Nn

## Homework Equations

i' = icosθ + jsinθ (3.3.20)

j' = -isinθ + jcosθ (3.3.21)

(x,y) -> (x',y')

and

r = xi + yj -> r = x'i' + y'j'

where

x' = xcosθ + ysinθ ,

y’ = xsinθ - ycosθ ,

i' = icosθ + jsinθ ,

and

j' = -isinθ + jcosθ

## The Attempt at a Solution

I can see how the problem works, but, if you click on the link to see the attached photos, I'm having more trouble with their derivations. I believe that I can see how the i' and j' were derived. I attached what I attempted as a photo at the end of the gallery. I do not understand how, in their derivation, they arrived at

x' = xcosθ + ysinθ and y’ = xsinθ - ycosθ. More specifically, it is the negative ycos that troubles me. The way I saw it was using the prior derivations and then finally grouping the terms that matched with i' and j', you would get y' = -xsinθ + ycosθ.In the given problem, the Aycos is negative, which makes more sense to me. I really would like to understand the underpinnings of this because it's crucial moving forward. Thank you so much.