Finding 4 perpendicular unit vectors The question:
Find 4 perpendicular unit vectors whose components are all either 1/2 or -1/2.
I' m just having trouble understanding how to do this.
NOTE:
The dimension is not specified.
 A: One vector could be $$\left(\begin{matrix}\frac 12\\\frac 12\\\frac 12\\\frac 12\end {matrix}\right)$$
Then the others can be 
$$\left(\begin{matrix}\frac 12\\\frac 12\\-\frac 12\\-\frac 12\end {matrix}\right)$$$$\left(\begin{matrix}\frac 12\\-\frac 12\\\frac 12\\-\frac 12\end {matrix}\right)$$$$\left(\begin{matrix}\frac 12\\-\frac 12\\-\frac 12\\\frac 12\end {matrix}\right)$$
A: First, deduce the number of entries. If a unit-vector has $n$ entries which are all $\pm 1/2$, what is $n$? Use the "Pythagorean theorem".
You should find that $n=4$. There are only $2^4=16$ vectors that you can work with, you just need $4$ that are perpendicular to each other. "Trial and error" is a reasonable way to go here.
A: I assume that the problem is in $\mathbb{R}^4$. Vectors $v,w \in \mathbb{R}^n$ are perpendicular if and only if their inner product equals zero. That is what you should use to solve this. You can just work with vectors whose entries are either $1$ or $-1$ since multiplication by a scalar does not affect orthogonality.
If you can't come up with a solution, you can Google something called "a Hadamard matrix" and you'll find one.
