Vectors/ Calculus with i and j components

WhiteWolf98

1. The problem statement, all variables and given/known data

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.

WhiteWolf98

Would I have to differentiate the position vector...?

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WhiteWolf98

I got: $$v=(\frac 1 2t^3-4\lambda t) \mathbf i + (10t-\lambda) \mathbf j$$

WhiteWolf98

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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2018 Award
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.

"Vectors/ Calculus with i and j components"

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