Parallel Vectors

[powerless]

Let a and b be non-zero parallel vectorsand let the points A,B,C and D be given by OA=a, OB=b, OC= 3a-b,OD = -a+2b

a) express the vectors AB,AC,AD,BC,BD and CD in terms of a and b.

b) Find, in terms of a and b, vectors from the origin to the midpoints of the line segments AB and CD.




3. The attempt at a solution

a) AB = b-a
AC = (3a-b)-a = 2a-b
AD = (-a+2b)-a = -2a+2b
BC = (3a-b)-b = 3a-2b
BD = (-a+2b)-b = -a+b
(-a+2b)-(3a-b) = -4a+b

Hope it's right.

b) I don't know how to do question. I appreciate it if someone could give me some guidance and show me how to solve this question.
Thanks!

[danago]

Let P be midpoint of the line AB. You have already found that AB = b-a, so what will AP be, knowing that AP is parallel to AB and that the magnitude of AP is half of the magnitude of AB?

Once you have found the vector from A to P, you should be able to find the vector from the origin to P, since you know the position vector of A.

[powerless]

AB = AO + OB
AB = -OA + OB
AB = -a + b

CD = CO + OD
CD = -OC + OD
CD = -(3a-b) + (-a + 2b)
CD = -3a + b -a + 2b
CD = -4a + 3b

midpoint of AB = (-a + b)/2

midpoint of CD = (-4a + 3b)/2

Is that correct?

[HallsofIvy]

AB = AO + OB
AB = -OA + OB
Surely you can't mean both of these! It has already been confirmed that AB= OB-OA, NOT AB= OB+OA.

AB = -a + b

CD = CO + OD
CD = -OC + OD
Same thing. The second equation is true, not the first.

CD = -(3a-b) + (-a + 2b)
CD = -3a + b -a + 2b
CD = -4a + 3b

midpoint of AB = (-a + b)/2

midpoint of CD = (-4a + 3b)/2

Is that correct?

[HallsofIvy]

Surely you can't mean both of these! It has already been confirmed that AB= OB-OA, NOT AB= OB+OA.


Same thing. The second equation is true, not the first.

By the way, I started to respond earlier but became terribly confused: your title says parallel vectors but your problem is about NON-parallel vectors!