Question

Suppose q1, q2, a3 are linearly independent vectors. q1 and q2 are already orthonormal. Give a formula for a third orthonormal vector
q3 as a linear combination of q1, q2, a3.
I have no idea where to even start with this one

Answer 1

Are we in real or complex?

Answer 2

real

Answer 3

You'd use Gram-Schmidt process, which for this case where q1 and q2 are orthonormal
x3=a3 -<a3,q1>q1 -<a3,q2>q2
After that, you'd normalize x3 to become q3.

Answer 4

I didnt even think about that thanks! So then given that, the answer Ive come up with is Q3=(a3-Q1Q1^Ta3-Q2Q2^Ta3)/II " " II

Answer 5

Yep, something along those lines, not too sure about your notations.
I forgot to mention <a3,q1> is the inner product, and for real, it's simply the dot-product.