A moving asteroid

[next wave]

Hi I don't know how to start on this one.

Translation:

An asteroid moves with constant speed v = (1, 3, 2) and is at r0 = (−1, −2, 0) when t = 0. Choose a coordinate system with the Earth as the origin and Longitude equal to the distance between the Earth and the Moon (384 400km). Time is further measured in days.
i. Determine an expression for the position of the asteroid at an arbitrary time e.g.

Should I use r(t) = r0 + v*t
r(t) = (-1, -2, 0) + (1, 3, 2)t. ?

(-1+t, -2+3t, 2t) ?

[ember atlas]

Yes, the equation is correct.

[next wave]

okay thanks,

I have 2 more follow up questions but lets start with ii)
"Where is the asteroid when it is closest to Earth?"

Im wondering if I should do something like this ((1, 3, 2) - (-1, -2, 0) = (2, 5, 2) ? I feel like its the wrong approach though,

[placid mortar]

@next wave has given 1 rep to @ember atlas

[ember atlas]

Oh, basically, you just need to find the distance from this line to the origin.

Oh, actually, it asks for position... Hm.

[ember atlas]

Well, not a problem. Let's find the formula.
Consider the squared distance from the origin as a function of t:
f(t) = r^2 = (r0 + st)^2
Let's find its minimum value. For that, we find f'(t) first.
f'(t) = 2(r0 + st)·s = 2(r0·s + |s|^2t)
And now we find where it's equal to zero.
2(r0·s + |s|^2t) = 0
t = -r0·s/|s|^2
That is the moment at which the asteroid is closest to Earth. To find the position, we just substitute that into r(t).
r(t) = r0 - (r0·s/|s|^2)s
This formula should also probably remind you of something.

[next wave]

r(t) = = (−1, −2, 0) - (−1, −2, 0)s /|s|^2 )s ?

what is s? the difference in position ?

[next wave]

I calculated the dot product instead to find the scalar product, so this is my D = (-1+t , -2+3t , 2t) and now take D dot D and figure out t

Is this method correct?

[ember atlas]

s is the direction vector. Though, yeah, I guess it would be better to denote it by v.
r0 = {-1, -2, 0}
v = {1, 3, 2}
r0·v = -7
|v|^2 = 14
r0·v/|v|^2 = -1/2
r0 - (r0·v/|v|^2)v = r0 + (1/2)v = {-1, -2, 0} + (1/2){1, 3, 2} = {-1/2, -1/2, 1}

[next wave]

thanks !