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

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The discussion addresses the confusion around calculating the velocities of point A in a system with two rods at fixed angles. It highlights that for part (a), the velocity of A cannot simply be expressed as v_A = -v_1 j hat due to the constraints of the system. In part (b), the velocity must account for the simultaneous movement along both rods, leading to a more complex expression than basic vector decomposition. The additional horizontal component in the solution for part (a) is necessary to satisfy the direction of motion along rod l2. Understanding the constraints of the system is crucial for accurate velocity calculations.
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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.

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Screenshot 2024-10-26 at 5.47.38 PM.png
 
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For a), A must move along the direction of l2.
For b), A must move along the direction of l2 and l1 simultaneously.
 
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