Vectors/ Calculus with i and j components

[WhiteWolf98]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

 

[haruspex]

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...?

 

[haruspex]

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

Yes.

 

[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]

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.