Extend Vector to Orthogonal Basis

[special-g]

How do you extend a vector (let's use vector (1,2,3) for example) to an orthogonal basis for R^3?

 

[Dick]

How do you extend a vector (let's use vector (1,2,3) for example) to an orthogonal basis for R^3?

First find a vector perpendicular to (1,2,3). That's not hard if you think about the dot product. Any one, there are many. Then find a vector perpendicular to the first two. That's easy in R^3 if you use the cross product. There is no unique answer, you know.

 

[special-g]

Thanks, that helps quite a bit...we just started this material and there aren't very many examples in the book to go off of. :)

I don't know if I did this completely right because I haven't taken calc 3 in a while.

For the first vector, would this be one answer: vector (-5, 1, 1) after taking the dot product.
Then, after taking the cross product of those two vectors, I got vector (-1,-16,11).