Vectors in rotated coordinate systems

Question

I’m sure it’s super simple, and I usually don’t have problems with trig, but for some reason I can’t wrap my head around converting between a vector in one coordinate system to one rotated about the origin with respect to that 1st coordinate system.

Specifically, I mean that (given the unit vectors in the original coordinate system to be **i** and **j**, and the second coordinate system to be **i’** and **j’,** and the angle to be Φ) we have:

**i’** = **i** cos(Φ) + **j** sin(Φ)

and

**j’** = -**i** cos(Φ) + **j** sin(Φ)

*not entirely sure if I have that second one correct

In: Physics

Answer 1

You’re just making the new i and j the hypotenuse (of the right triangle with that angle) and scaling back to unit magnitude.