1. ## Vector question

If $A(-1,3,4), B(4,6,3), C(-1,2,1)$ and $D$ are the vertices of a parallelogram, find all the possible coordinates for the point D

I managed to find one point using the fact that the vector AB is parallel to the vector CD and hence finding the vector of D. However, I cannot find the remaining two points.

2. D is adjacent to either A and B, B and C, or A and C.

If D is adjacent to A and B, then it is opposite C. You can see that in this case D = C + (A-C) + (B-C) = A+B-C by drawing a picture.

Permuting the roles of A, B and C gives all possibilities.

D is adjacent to either A and B, B and C, or A and C.

If D is adjacent to A and B, then it is opposite C. You can see that in this case D = C + (A-C) + (B-C) = A+B-C by drawing a picture.

Permuting the roles of A, B and C gives all possibilities.
Sorry, I still don't exactly understand.

4. Originally Posted by acevipa
Sorry, I still don't exactly understand.
One of the possible points is given by $A+B-C$, as indicated above. However, it doesn't matter how you order $A,B,C$ in that expression. So you can have $A+C-B$ and $B+C-A$ as other possible points, exhausting possibilities.

5. Originally Posted by hatsoff
One of the possible points is given by $A+B-C$, as indicated above. However, it doesn't matter how you order $A,B,C$ in that expression. So you can have $A+C-B$ and $B+C-A$ as other possible points, exhausting possibilities.
Yes but how do you get that? I mean don't we label points in order. So if we have a parallelogram ABCD:

Isn't $AB || CD$ and $BC || AD$

So wouldn't we get:

$\vec{AB}=\lambda\vec{CD}$

$\vec{b} - \vec{a}=\vec{d}-\vec{c}$

$\Rightarrow\vec{b} +\vec{c} - \vec{a}=\vec{d}$

$\vec{BC}=\lambda\vec{AD}$

$\vec{c} - \vec{b}=\vec{d}-\vec{a}$

$\Rightarrow\vec{c} +\vec{a} - \vec{b}=\vec{d}$

6. Hello acevipa
Originally Posted by acevipa
Yes but how do you get that? I mean don't we label points in order. So if we have a parallelogram ABCD:

Isn't $AB || CD$ and $BC || AD$
Yes, provided we are told that the parallelogram is ABCD. However, if we're just told (as we were in your original phrasing of the question) that A, B, C and D are the vertices of a parallelogram, then the parallelogram could also be ABDC or ACBD. So there are, as has already been said, three possible positions for D.