I'm working through some examples in Strang (Ed. 3) and the solution manual.
A couple questions in a row now have been "find vectors perpendicular to the given vector as well as perpendicular to each other"
The 3d one I solved by drawing out a 3-d matrix. The original vector was (1,0,1). I visually draw one perpendicular to that (-1,0,1). And then visually inspect to see the next perpendicular vector I can see (0,1,0)
Then I was given a 4d example -- find three vectors perpendicular to (1,1,1,1) and each other.
In this case I have a hard time drawing it. What's the best way to approach a solution for this kind of problem (nothing stopping the text from asking me for a 6-dimension example next)? Should I just find an arbitrary perpendicular vector. And then for the second vector work out multiple equations for (A.v2) and (v1.v2) and solve for the v2 parameters? and then keep doing that till I can generate a v3? Or should I be smarter in my intuition here.
Feeling dumb. I don't think I have a foolproof approach to this problem class.