A particle has position vector 2i + j initially and is moving with speed 10m/s in the direction 3i - 4j. Find its position vector when t = 3 and the distance it has travelled in those 3 seconds.

My solution:

Position vector = initial position + displacement

since displacement = velocity x time

then position vector= (2i + j) + 3(3i - 4j)

= 2i + j + 9i - 12i

= 11i - 11j

and distance is straightforward, distance = speed x time so 10 x 3 which is 30m.

My solution is however wrong, becuase in the question it says (3i - 4j) is the direction the particle is travelling in, not the velocity, so im kinda stuck and the right answer should be 20i - 23j according to the back of the book.

How do i do this question? Also, if u find the velocity vector, and work out the magnitude, you should get the speed right? So if initial velocity is (2i + 3j) then the magnitude of this vector should equal the speed which is 10 m/s but it does not, the magnitude of this vector is the sqrt (2^2 + 3^2) which is root 13 and not 10 m/s so how does that work? Thanks for you help.