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 $

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- Jan 27th 2008, 10:15 AMtttcomraderFind 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 $

- Jan 27th 2008, 11:01 AMSoroban
Hello, tttcomrader!

I don't understand the problem . . .

Quote:

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.

- Feb 2nd 2008, 06:02 AMtttcomrader
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. - Feb 2nd 2008, 08:53 AMSoroban
Hello, tttcomrader!

It still has unanswered questions . . .

Quote:

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?

- Feb 2nd 2008, 02:07 PMtttcomrader
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...