Find four perpendicular unit vectors?

Question

Find four perpendicular unit vectors with all components equal to 1/2 or -1/2.

Answer 1

Clearly, these have to be 4-vectors, or they couldn't be all perpendicular to each other. So they must be of the form [±½,±½,±½,±½]. You want all of their dot products to be zero, which means you have to have two +½'s and two -½'s. This leaves us with

[½,½,-½,-½]

[½,-½,½,-½]

[-½,½,½,-½]

[-½,-½,½,½]

Answer 2

In a normal trig unit vector circle the length is one, and at 45 degrees the x and y components are all sqrt(2)/2 or -sqrt(2)/2.

 

In this case, if we make the unit circle radius 1/sqrt(2), then the vectors at 45 degrees to the cartesian axes have x and y components of 1/2 or -1/2 depending on which quadrant they are in.