M1 Vectors help needed (Edexcel)

If B is due north of A the i components of your position vectors are equal

Therefore: 6-t = -3+t so t=4.5

So we can say 4.5 hours after noon (12:00) is 4:30pm

Why are they equal?

I'm having difficulty understanding the answer to Mixed Ex 6H Q2 from the Heinemann textbook.

"A coastguard station O monitors the movements of ships in a channel. At noon, radar records two ships with constant speed. A: r0 = (-3i +10j)km relative to O and v = (i + j) km/h. B: r0 = (6i + j)km rel. O and v = (-i + 5j)km/h. Find the time at which B is due north of A. (i = East, j = north).

Here's my working:

rA = (-3 + T)i + (10 + T)j
rB = (6 - T)i + (1 + 5T)j

If B north of A, j(B) > j(A) therefore:

1 + 5T > 10 + T
4T > 9
T > 9/4, which equals 2h15mins therefore time = 2:15pm

However, the book says it is 4:30pm and the solution bank online says you need to add up the "i"s to find when B is due north of A. Is the textbook wrong?

Think about the positions as coordinates, where the i components are along the x-axis and the j-components are along the y-axis.

If you have one coordinate that is due north of another, the x-coordinates would be equal wouldnt they?


I hope I am making sense

The books working will look like (-3+T) = (6-T) , as due north when delta i = 0

2T=9

therefore T =4.5 hours after noon, i.e. 4.30 pm