1. ## Quick elementary question regarding vectors

The below picture is actually a top-down view of a 3d cube. The corners on the cube are labelled points A, B, C, and D.

My question is: If I know that the vector AC = (-11, 26, 8) and BD = (-26, 11, 0), how can I find the 3d coordinates for point D? In other words: B = (x, y, 0), find x and y.

I know it involves halfing both vectors to obtain the midpoint of CA and BD, just not sure how to use that information to find x and y.

EDIT: If helpful, point C = (2, 4, 17).

2. Remember the magnitude (length) of a vector is:

$\displaystyle l = \sqrt{x_1^2 + x_2^2 + ... + x_n^2}$

Where $\displaystyle x_1^2 + x_2^2 + ... + x_n^2$ are the number of coordinates you have.

In this case we have 3. So the length or $\displaystyle AC = \sqrt{x^2 + y^2 + z^2}$

By the way, did you come up with C = (2, 4 , 17) on your own? Or was that a given? Also, is it specific where the points are on the cube? From that picture they all look to be on the same side, but that might not be true.

3. Originally Posted by eXist
Remember the magnitude (length) of a vector is:

$\displaystyle l = \sqrt{x_1^2 + x_2^2 + ... + x_n^2}$

Where $\displaystyle x_1^2 + x_2^2 + ... + x_n^2$ are the number of coordinates you have.

In this case we have 3. So the length or $\displaystyle AC = \sqrt{x^2 + y^2 + z^2}$

By the way, did you come up with C = (2, 4 , 17) on your own? Or was that a given? Also, is it specific where the points are on the cube? From that picture they all look to be on the same side, but that might not be true.
Point C was given.

Yes I'm aware of the length formula, but I don't think that is helpful for finding the coordinates of D.

4. To me, it seems like there are 2 answers because you are un-aware of whether C and D like on the same side or not. You can find the dimensions of the cube fairly easy by finding the length of $\displaystyle BD = \sqrt{(-26)^2 + (11)^2} = \sqrt{797}$ this gives you the length of the diagonal of one side.

Since the triangle that is made by dividing the side of a cube by a diagonal is a 45/45/90 triangle, we know that the length of cube = $\displaystyle \sqrt{797}/\sqrt{2} = \sqrt{797/2}$

Now that we have the length of a side, we don't know whether D lies above or below C.

5. Originally Posted by eXist
To me, it seems like there are 2 answers because you are un-aware of whether C and D like on the same side or not. You can find the dimensions of the cube fairly easy by finding the length of $\displaystyle BD = \sqrt{(-26)^2 + (11)^2} = \sqrt{797}$ this gives you the length of the diagonal of one side.

Since the triangle that is made by dividing the side of a cube by a diagonal is a 45/45/90 triangle, we know that the length of cube = $\displaystyle \sqrt{797}/\sqrt{2} = \sqrt{797/2}$

Now that we have the length of a side, we don't know whether D lies above or below C.
D lies below C, it is given in the problem.

6. Originally Posted by Ares_D1
D lies below C, it is given in the problem.
This helps, but our work isn't done. We need to find some more points on the cube. How can we find A with the info that we have?

7. Originally Posted by eXist
This helps, but our work isn't done. We need to find some more points on the cube. How can we find A with the info that we have?
A is given too lol. Sorry, didn't realize we would need it.

A = (13, 30, 9) (I've rounded the coordinates for simplicity, shouldn't matter for now, just trying to figure out the method rather than the actual answer).

8. Well now that we know 2 corners of the cube, this determines the cube. Therefore we can find the other points.