# Linear Algebra finding points with certain conditions

1. a) Consider an equilateral triangle in the i-j-plane with one vertex at the origin, and a second vertex with position vector i. Let $u = \textit{a}\textbf{i}+\textit{b}\textbf{j}$ be the position vector of the third vertex of the equilateral triangle in the positive quadrant. Find a and b by requiring that the side lengths of the triangle are equal. (There are many ways to find this point, in preparation for part (d) use vector algebra and the formula for the length of a vector to do this).

b) In addition to the three vertices of the equilateral triangle in the i-j plane found in (a), show that a fourth vertex with position vector $v = 1/2 \textbf{i} + \sqrt{1/12}\textbf{j} + \sqrt{2/3} \textbf{k}$ has distance one to each vertex of the equilateral triangle from part (a).

c) The four points from part (a) and (b) are the vertices of a regular tetrahedron. Consider the face of the tetrahedron that does not contain the origin. Show that the plane that contains this face is the plane $\sqrt{6)x + \sqrt{2}y + z = \sqrt{6}$

d) In addition to the four vertices of the tetrahedron in the i-j-k space from part (a) and (b) nd the position vector $w = \textit{a}\textbf{i}+\textit{b}\textbf{j}+\textit{c}\textbf{k}+\textit{d}\textbf{l}$ of a fifth point in 4-dimensional space with all components positive that has distance one to each vertex of the tetrahedron.

My attempt at this question:

a) $a = 1/2$ and $b = \sqrt{3}/2$

b) I just computed the distances and checked that they were all of length 1

c) I subbed in the points of the tetrahedron that weren't the origin and since they satisfied the plane, the face of the tetrahedron that made from those points were contained in the plane.

Could someone please verify these for me and give me some hints on part (d). I thought it was as simple as finding the four lengths and solving for $\textit{a, b, c, d}$ but I kept getting $\textit{d^2}$ to be negative. Thanks for any help!

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I understand that my mathematical symbols are not correct. Could someone please fix them for me so that the questions is easier to read for all and please let me know how to fix them so I can get it right in future questions/answers. Thanks! – Joe S May 5 '13 at 7:19
Just put a dollar sign on either side of each formula and it should work fine. Click the MathJax link in the right sidebar in edit mode for more information. – Daniel McLaury May 5 '13 at 7:22