Finding Orthogonal Vectors in R^5 without Cross Product

[phil ess]

Homework Statement

Given vectors a=(1,-1,0,2,1) b=(3,1,-2,-1,0) and c=(1,5,2,4,-4), which are mutually orthogonal, find a system of linear equations that a vector x must satisfy so it is orthogonal to a, b, and c.

Homework Equations

None I think.

The Attempt at a Solution

This is part A of the problem, after which I just have to solve the system and I'm confident I can do the rest once I have it, but I don't know where to start. I tried using the equations:

x1-x2+3x4+x5=0
3x1+x2-2x3-x5=0
x1+5x2+2x3+4x4-4x5=0

And then just solving the matrix, which gives me
x=(0,0,0,0,0)+s(-8,2,-11,5,0)+t(-5,5,-6,0,2)

But I'm pretty sure this doesn't answer the problem. I don't know how to create a system that gives orthogonal solutions, and the above solution isn't even a vector as far as I know. How do I start this problem? Any tips are much appreciated! Thanks.

 

[Dick]

You are doing exactly the right thing, except there seems to be some typos in your equations. How did you get the 3 in the first one and why is there an x5 and not an x4 in the second one. Your result isn't 'a' vector, it's a two parameter family of vectors, which is exactly what you should get, but the numbers aren't quite right either.

 

[phil ess]

I understand the problem now, and I went back and reworked everything carefully and ended up with the right solution. It was actually much easier than I thought at first. Thanks a lot!