Linear Algebra - Find unit vector orthogonal to 2, 4-space vectors?

[twotaileddemon]

yes yes, you are right once again!

I'm sorry, I make lots of mistakes at 4:40 am xD
I wanted to do this homework so badly so I must get it done!

 

[angryfaceofdr]

its fine, happens to everyone

 

[HallsofIvy]

In order to solve for specific values of a, b, c, and d, you would need four equations. But the problem said "Find any unit vector orthogonal to both of them". Just as, in three dimensions, there are an infinite number of unit vectors orthogonal to a single vector, in four dimensions there are an infinite number of unit vectors orthogonal to two vectors.

In order that (a, b, c, d) be orthogonal to u = (2, 0, 1, -4) and v = (2, 3, 0, 1) we must have, as you say, 2a+ c- 4d= 0 and 2a+ 3b+ d= 0. Subtracting the first equation from the second, 3b- c+ 5d= 0 so c= 3b+ 5d. Put that back into either of the first two equations and you can solve for a in terms of b and d. That gives every vector that is orthogonal to u and v in terms of b and d.