Suppose a vector $a$ is given. What is precisely meant by projecting the component of vector $b$ out of $a$? Does that mean that the resulting vector $a_1$ (obtained by "projecting out the component of $b$ from $a$") is orthogonal to $b$? How could one achieve this?
I read that Gram-Schmidt process is used for this; however, it involves the notion of "normalization", and I'm not sure if that means that the vector $a_1$ would be normalized.