Two rods with fixed angle, find the velocity of the crossing point

[mancity]

Homework Statement

Two straight rods l1 and l2 are put on a piece of paper as shown in the figure. the angle between two rods is theta. What is the velocity of the crossing point A when l1 starts to roll with velocity v1? What about if l2 also starts to roll with velocity v2?

Relevant Equations

Vector decomposition

I apologize if this is a stupid question but how come we can't just say that for part (a) v_A = -v_1 j hat, and for (b) v_A=v2 sintheta i hat + (-v1-v2cos theta) j hat? i.e. how come we can't just do vector decomposition "normally"? I am especially confused about the given solution in (a) where there is an additional horizontal component of velocity in the i directioneven though v1 appears to only act in the j direction.

Thanks

 

[Lnewqban]

For a), A must move along the direction of l₂.
For b), A must move along the direction of l₂ and l₁ simultaneously.