How can you orthogonalize a set of vectors

Question

I have been reading "Mathematics for 3D Game Programming and Computer Graphics" and there is a chapter exercise (Chapter 2. Question 2) that despite rereading the chapter and researching, I can not seem to understand. How can I "Orthogonalize the following set of vectors"

e1 = ( sqrt(2)/2, sqrt(2)/2, 0 )

e2 = ( -1, 1, -1 )

e3 = ( 0, -2, -2 )

Also, what does it mean to "Orthogonalize a set of vectors"?

Answer 1

The Gram-Schmidt Process is the typical method used to derive an orthonormal basis for the spanned space defined by a collection of linearly independent vectors. In the case you describe, since e1, e2 and e3 are linearly independent, Gram-Schmidt can be used to generate three mutually orthogonal vectors of unit length e1', e2' and e3' which is an orthonormal basis of the linear span of your original vectors.