3d proofs

[Clifford]

We have now started into 3d geometry and our professor told us to start thinking about coordinate proofs.

What kind of questions can you prove cubes, rectangle prisms and such?

I know for 2d you can prove slopes, length of lines midpoints and stuff, but is that possible with 3d because you have an extra dimension?

 

[pka]

We have now started into 3d geometry and our professor told us to start thinking about coordinate proofs. I know for 2d you can prove slopes, length of lines midpoints and stuff, but is that possible with 3d because you have an extra dimension?

All of the above will occur in \(\displaystyle R^3\) except slope is replaced by direction vectors. You will have equations of planes. \(\displaystyle R^2\) is just a subspace of \(\displaystyle R^3\)

 

[Clifford]

so say if we plot a cube on a grid we could get points
O(0,0,0)
M(0,x,0)
N(x,x,0)
P(x,0,0)
Q(0,x,x)
R(x,x,x)
S(0,0,x)
T(x,0,x)

say if I wanted to calculate the point at which all the diagonals intersected. By knowing its a cube you would know that they all would intersect at (.5x,.5x,.5x). What would the mathetimatical proof for this be?

I was thinking to get the equations of two lines.
l1 = (0,0,0) + t(x,x,x)
l2 = (0,0,a) + k(x,x,-x)
with actual vectors (with numbers), we could break these into parametric equations
get a value for t and k and sub back into the parametric equations to get the point.

Would this be the right approach for this type of thing or should I go about it another way?