Hello everyone, I'm new here, so please do not make fun of me if this is a seemingly stupid question because I have almost no idea how to go about it.

Problem:

Consider these three orthonormal vectors in R^5:

v1 = 1/3[2 2 1 0 0], v2 = 1/7[0 3 -6 2 0], v3 = [0 0 0 0 1]

Write the following vectors in R^5 as linear combinations of these three vectors plus a vector orthogonal to all three. (If the vector is in the subspace that has these three vectors as its basis, the orthogonal "extra vector" will be all zeros.]

[9 4 6 0 6] = _______________________ + remainder vector: ___________

I have attempted the problem because I understand the portion in which he states "write... as linear combinations..." in which I simple dotted the given vector with v1, v2, and v3, respectively:

w . v1 (vector w dot vector v1) = 32/3
w . v2 = -24/7
w . v3 = 6

So, w = (32/3)u1 + (-24/7)u3 + 6u3.

After this, I'm lost, however, it should be noted that in class, he did something similar to this, but he used a funky interpretation of the Gram-Schdmit Process that I simply couldn't follow it, perhaps, that might be the solution, but I haven't gotten it to work.

Thanks everyone, please give me input ASAP since my assignment is due tonight at 5 PM PST, and this is the only problem I'm stuck on. Forgive me for not using Latex as I do not have it nor I do I know its syntax.