I've been trying to solve this question all day. If somebody could point me in the right direction I would really appreciate it! (ii) A particle’s motion is described by the following position vector r(t) = 4txˆ + (10t − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r.  v(t) is given as: v(t) = 4xˆ + (10t − 1)ˆy (iv) Using the ˆr and ˆθ you found in (ii) above, write v(t) as a linear combination of rˆ and ˆθ.  (v) Differentiate the expression for r(t) you got in part (ii) (in terms of ˆr and ˆθ, and using the expressions ˙rˆ = ˙θ ˆθ , ˙ˆθ = − ˙θrˆ derived in the lectures, show that you obtain the same answer as in part (iv) I understand that for θ^ = -sinθ + cosθ and r^=sinθ + cos θ I'm just unsure how to do part (ii) without it being really messy e.g with cos(tan^-1(10t-1/4)) where tan^-1(10t-1/4)=θ especially knowing I have to find a linear combination afterwords. The attempt at a solution My current answer for part (ii): r^= cos(tan^-1(10t-1/4))x^ + sin(tan^-1(10t-1/4))y^ θ^= -sin(tan^-1(10t-1/4))x^ + cos(tan^-1(10t-1/4))y^ And for part (iv) am I am right in saying that I need to find, a and b scalars for the following: 4 = a*cos(tan^-1(10t-1/4)) +b*-sin(tan^-1(10t-1/4)) 10t-1 = a*sin(tan^-1(10t-1/4))+b*cos(tan^-1(10t-1/4)) I'm really not sure about this one. It appears like it's going to difficult to eliminate the t value. Anyway, thanks in advance, I'm going to keep working on it now.