AS MECHANICS-kinematics in two direction question, help needed, thank you

A boat moves so that its position in metres at time t seconds is given by r=(2t-6)i+(3t-9)j where i and j are unit vectors that are directed east and north respectively.
A lighthouse has position 186i+281j.
Calculate the times when the boat is due south of the lighthouse and the time when the boat is due east of the lighthouse.

please explain in steps

thank you

When the boat is directly south of the lighthouse, the vector from lighthouse to the boat is pointing south, but the magnitude is not known. In general any vector pointing south is:

kj

where k should be a negative real number. Also the vector from the lighthouse to the boat at any given time is just the vector to the boat minus vector to lighthouse:

[(2t-6)i+(3t-9)j] - [186i+281j]
= (2t - 192)i + (3t - 290)j

so when the boat is directly south of the lighthouse, this vector should equal kj :

(2t - 192)i + (3t - 290)j = kj

the two scalar equations are:

2t - 192 = 0
t = 96

and

3t - 290 = k
3*96 - 290 = k
k = -2

since k is negative, the vector from lighthouse to boat is -2j. If k was positive, the vector would be pointing north and the answer would be extraneous.

for east the vector would be ki (k should be positive). Do the same thing again:

(2t - 192)i + (3t - 290)j = ki

3t - 290 = 0
t = 290/3

2t - 192 = k
k = 580/3 - 192
k = 4/3

again, if k was negative, it would be west of the lighthouse and the answer would have been extraneous.

The position vector of the boat from the lighthouse is
r - (186i + 281j)
= (2t-6)i+(3t-9)j - 186i - 281j
= (2t - 192)i + (3t - 290)j

The boat is due south of the lighthouse when this position vector has zero east component,
i.e. 2t - 192 = 0
t = 96
So it's due south after 96 second.
Note that at that instant, 3t - 290 = -2, confirming that it's south, not north. (south negative, north positive)

It's due east when the north component is 0,
i.e. 3t - 290 = 0
t = 290/3 = 96.6666....
i.e. after 96⅔ second.
At that time, 2t - 192 = 1⅔, confirming that it's east, not west.

The boat is due south when the "i" coordinate of the boat & the lighthouse coincide:
2t -6 = 186 ==> t = 96 s.
Similarly, the boat is due east when, 281 = 3t -9 ==> t = 290/3 s.