Find vectors

• January 27th 2008, 10:15 AM
Find vectors
Find 4 vectors $V_{1},V_{2},V_{3},V_{4}$ such that $|V_{i}| = 1 \ for \ i=1,2,3,4, (V_{i},V_{j})=0, i \neq j$
• January 27th 2008, 11:01 AM
Soroban

I don't understand the problem . . .

Quote:

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

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

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

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

• February 2nd 2008, 06:02 AM
I asked the professor and I finally understand what he was writing, the correct question is:

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

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.
• February 2nd 2008, 08:53 AM
Soroban

It still has unanswered questions . . .

Quote:

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

and: . $(V_iV_j)\:=\:e,\;\;i \neq j$. . I still don't know what ${\color{blue}(V_iV_j)}$ 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: . $A(0,0,0),\;B(1,1,0),\;C(0,1,1),\;D(1,0,1),\;\;(1,0 ,0),\;(0,1,0),\;(0,0,1),\;(1,1,1)$

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

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

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

Is there an original wording of the problem?

• February 2nd 2008, 02:07 PM
It is dot product. $e_{1} = (1,0,0)$