How to prove that the sum of two rotating vectors in an ellipse?

[ppoonamk]

Homework Statement

Within the xy-plane, two vectors having lengths P and Q rotate around the z-axis with angular velocities ω and –ω. At t = 0,these vectors have orientations with respect to the x-axis specified by θ1 and θ2. How do I find the orientation of the major axis of the resulting ellipse relative to the x-axis.

The Attempt at a Solution

P=|p|cos(θ1+ωt) x^+ |p|sin(θ1+ωt) y^
Q=|q|cos(θ1+ωt) x^+ |q|sin(θ1+ωt) y^

x^- x hat
y^-y hat

How do I solve this after these 2 equations?
I tried to group the x and y vectors separately. But i could not figure out anything after that.

X= |p|cos(θ1+ωt)+|q|cos(θ1+ωt)
Y=|p|sin(θ1+ωt)+ |q|sin(θ1+ωt)

Without the θ terms I could have just squared both sides and added it. But now I am stuck. Thank you for the help :)

 

[ehild]

Homework Statement

Within the xy-plane, two vectors having lengths P and Q rotate around the z-axis with angular velocities ω and –ω. At t = 0,these vectors have orientations with respect to the x-axis specified by θ1 and θ2. How do I find the orientation of the major axis of the resulting ellipse relative to the x-axis.

The Attempt at a Solution

P=|p|cos(θ1+ωt) x^+ |p|sin(θ1+ωt) y^
Q=|q|cos(θ1+ωt) x^+ |q|sin(θ1+ωt) y^

x^- x hat
y^-y hat

How do I solve this after these 2 equations?
I tried to group the x and y vectors separately. But i could not figure out anything after that.

X= |p|cos(θ1+ωt)+|q|cos(θ1+ωt)
Y=|p|sin(θ1+ωt)+ |q|sin(θ1+ωt)

Without the θ terms I could have just squared both sides and added it. But now I am stuck. Thank you for the help :)

The expressions are not correct.
The vectors have length P and Q, use them instead of |p| and |q|.
The vectors rotate in opposite directions (the angular velocities are ω and -ω).
One vector encloses θ1 angle with the x-axis at t=0, the other one encloses θ2.

Expand the cosine and sine terms in the expression for X and Y, collect the terms with cos(ωt) and sin(ωt).
There is an easy method to find the angle of the principal axis: Just think that the vectors rotate in opposite directions, and there is a time instant when they are on the same line, so the resultant has the longest length.
ehild