1. ## Find vectors

Find 4 vectors $\displaystyle V_{1},V_{2},V_{3},V_{4}$ such that $\displaystyle |V_{i}| = 1 \ for \ i=1,2,3,4, (V_{i},V_{j})=0, i \neq j$

I don't understand the problem . . .

Find 4 vectors: $\displaystyle V_{1},V_{2},V_{3},V_{4}\text{ such that }|V_{i}| = 1\text{ for }i=1,2,3,4$ . . . . They are unit vectors

and: .$\displaystyle (V_{i},V_{j})=0,\;i \neq j$ .
. . . What does this mean?

If you meant: .$\displaystyle V_i\cdot V_j \:=\:0$, we have four mutually perpendicular vectors.

This is possible in dimensions greater than or equal to 4.

3. I asked the professor and I finally understand what he was writing, the correct question is:

Find 4 vectors $\displaystyle V_{1},V_{2},V_{3},V_{4}$ such that $\displaystyle |V_{i}| = 1 \ for \ i=1,2,3,4, (V_{i}V_{j})=e, i \neq j$

He also gave me a hint:

Consider the unite cube and the regular tetrahedron which vertices are among the vertices of the cube. Connect the center of the cube with the vertices of the tetrahedron.

It still has unanswered questions . . .

Find 4 vectors $\displaystyle V_1,\:V_2,\:V_3,\:V_4$ such that: .$\displaystyle |V{i}| = 1$ for $\displaystyle i=1,2,3,4$ .(unit vectors)

and: .$\displaystyle (V_iV_j)\:=\:e,\;\;i \neq j$. . I still don't know what $\displaystyle {\color{blue}(V_iV_j)}$ means ... dot product?

And does it really equal e = 2.71828... ?

He also gave me a hint:
Consider the unit cube and the regular tetrahedron which vertices are
among the vertices of the cube. Connect the center of the cube with
the vertices of the tetrahedron. . Then what?

Place the cube in the first octant with one vertex at the origin.

The vertices are: .$\displaystyle A(0,0,0),\;B(1,1,0),\;C(0,1,1),\;D(1,0,1),\;\;(1,0 ,0),\;(0,1,0),\;(0,0,1),\;(1,1,1)$

$\displaystyle A,\,B,\,C,\,D$ are the vertices of a regular tetrahedron with side $\displaystyle \sqrt{2}$.

The center of the cube is: .$\displaystyle O\left(\frac{1}{2},\:\frac{1}{2},\:\frac{1}{2}\rig ht)$

Then: .$\displaystyle \begin{array}{ccccc}V_1 &=& OA &=& \langle \text{-}\frac{1}{2},\;\text{-}\frac{1}{2},\:\text{-}\frac{1}{2}\rangle \\ V_2 &=& OB&=&\langle \frac{1}{2},\:\frac{1}{2},\:\text{-}\frac{1}{2}\rangle \\ V_3 &=&OC &=&\langle\text{-}\frac{1}{2},\:\frac{1}{2},\:\frac{1}{2}\rangle \\ V_4 &=& OD &=& \langle \frac{1}{2},\:\text{-}\frac{1}{2},\:\frac{1}{2}\rangle\end{array}$ . . (These are not unit vectors.)

Is there an original wording of the problem?

5. It is dot product. $\displaystyle e_{1} = (1,0,0)$

I asked him how to do this, he explained to me a great deal, but I do not get any of it, as a matter of fact, I still don't understand what the question is asking. I don't think I'm a stupid person, after all I did finish my history BA and math BS with GPA over 3.6, but in this class, I"m completely lost no matter how hard I try...