Yes, everything you did was correct. My only question is why, having analysed this problem, done the calculations, the checked and seen that your answer worked, why do you not have confidence in it?
Given the points A(2, -3, 1) and B(8, 9, -5) in , find:
The coordinates d of point D on the line AB such that D is between A and B and
First I found the vector by subtracting A from B to get (6, 12, -6). By drawing a small diagram, I realised I had to find the co-ordinate vector 2/3 the way to B from A. So I wrote the parametric form of a line with origin A (using the vector that I calculated). I subbed in 2/3 as the scalar multiple, and produced (6, 5, -3) as the answer to this question.
I checked this by finding and equating it to and both were equal to (4, 8, -4). I'm fairly sure my answer is correct, but my text has no solutions for practise exams, so I'd like to be sure.