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Vectors/ Calculus with i and j components

1. The problem statement, all variables and given/known data
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I don't understand how to form an equation using the knowledge that, 'When ##t=4##, ##P## is moving parallel to the vector ##\mathbf {j}##'. I've seen the solution, and not a single part of it makes sense. I haven't attempted any question like this before, so I have no idea where to even start.

What I do know is how the whole integration/ differentiation process works to get between displacement, velocity and acceleration. That's only with normal equations though, not ##\mathbf {i}## and ##\mathbf {j}## components. So, I don't see any of what I do know already helping me here. Any help would be appreciated.
2. Relevant equations


3. The attempt at a solution
 

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haruspex

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how to form an equation using the knowledge that, 'When t=4, P is moving parallel to the vector ##\mathbf {j}##.
Can you write down the velocity vector? The i and j do not really create any complication for that. Just treat them as unknown constants.
 
Would I have to differentiate the position vector...?
 
I got: $$v=(\frac 1 2t^3-4\lambda t) \mathbf i + (10t-\lambda) \mathbf j$$
 
Uh, I don't think that this question can be solved so I'm going to close it; I don't believe that I have the ability yet. Thanks for the help anyway
 

haruspex

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I got: $$v=(\frac 1 2t^3-4\lambda t) \mathbf i + (10t-\lambda) \mathbf j$$
Good.
Next, you need the condition for this velocity vector to be parallel to j. That's easy: it just means the i factor is zero.
 

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