Projecting vectors from R3 onto a subspace

[Ghost of Progress]

I want to project a vector from R3 onto a subspace.
I'll let the bases for the subspace be [a,b,c]T
(my T's mean transpose)
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I have the definition for vector projection
p = (<u,v>/<v,v>)*v
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I know v will be the [a,b,c]T vector but what is u?
The only thing I could think of is let it be the triplet [x,y,z]T which could be any vector in R3.
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Using this I get
p = [(a/(a^2 + b^2 + c^2))(ax + by + cz)]
[(b/(a^2 + b^2 + c^2))(ax + by + cz)]
[(c/(a^2 + b^2 + c^2))(ax + by + cz)]
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I'm not very confident with this solution, I was hoping someone could tell me if this is correct or show me where I've gone wrong.

 

[matt grime]

u is presumably the vector you want to project onto the subspace.

 

[Ghost of Progress]

I was just working an a problem that askes me to find the projection matrix P that projects vectors in R3 onto the orthoginal compliment of a two dimensional subsapce of R3 spanned by
x1 = [1,0,2]T x2 = [0,1,-2]
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I've found that the bases of the orthoginal compliment is [-2,2,1]T
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Using the definition of P that a posted above I've found
P = [(-2/9)(-2x +2y +z)]
[ (2/9)(-2x +2y +z) ]
[ (1/9)(-2x +2y +z) ]

where, like you said, u = (x,y,z) = the vector I want to project onto the subspace. Does this seem like a correct solution?