Find an orthogonal basis.

Find an orthogonal basis for span(u,v,w) that includes u.

u = <3, -2, 3, 1>
v = <2, 3, -3, 5>
w = <17, -20, 27, -3>

I'm very lost here. Do I have to use the gram-schmidt process for all three of these vectors?
Any help provided is greatly appreciated!

Yes, you should use the gram-schmidt process.

Say you want to convert your u, v, w vectors to an orthogonal basis a, b and potentially c (it's unclear whether span(u, v, w) is a two or three dimensional subspace of R4).

Since your basis must include u, start with

a = u

b = v - [(v * a) / (a * a)] * a

c = w - [(w * b) / (b * b)] * b - [(w * a) / (a * a)] * a

What you'll find is that c returns a 0 vector. What this means is that w is already spanned by the vectors a and b, so w has no component orthogonal to that subspace.

So u, v, w span a plane in R4 and an orthogonal basis for that plane is the vectors a and b given above.

Yes, this is the gram-schmidt algorithm. Choose your first vector

b1 = u
=<3, -2, 3, 1>

Then, define b2 to be orthogonal to it.

b2 = v - (b1,v)/(b1,b1) b1.

Using the inner products

(b1,v) = <3, -2, 3, 1> * <2, 3, -3, 5>
=3*2-2*3-3*3+1*5
=-4

(b1,b1) = 3^2+(-2)^2+3^2+1^2
=23

we get

b2 = <2, 3, -3, 5> +4/23 <3, -2, 3, 1>
= 1/23*<58,61,-57,119>

Similarly, define b3 to be orthogonal to b1 and b2

b3 = w - (b1,w)/(b1,b1) b1 - (b2,w)/(b2,b2) b2

which gives

b3 = <17, -20, 27, -3> - 169/23*<3, -2, 3, 1> -2130 /24495*<58,61,-57,119>
= <0, 0, 0, 0>

This shows that the vectors are Linearly Dependent, and the first two vectors b1 and b2 form an orthogonal basis for Span{u,v,w}