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let $v$ be a vector, and $\forall i \in 1..k < n : n_i$ be normal vectors of hyperplanes in the in an $R^n$ space. the problem is to compute orthogonal Projection of $v$ onto intersection of all those planes.

I'm looking for an n*n matrix (M), so that $vM = Projection(v)$

Here are two examples: $$n=3, k=2, n_1=(1,0,0), n_2=(0,1,0) \Rightarrow p(1,1,1) = (0,0,1)$$$$n=3,k=2,n_1=(1,0,-1),n_2=(0,1,-1) \Rightarrow p(0,0,1)=({1 \over 3},{1 \over 3},{1 \over 3})$$$$n=3,k=1,n_1=(0,0,1) \Rightarrow p(1,1,1) = (1,1,0)$$

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    What do you mean by image of $V$? $V$ is a vector. What does $\times$ mean?2012-12-12
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    @copper.hat removed $\times$ I just meant normal multiplication function of matrices. here is an example let $n=3$ and $V =(1,1,1)$ and $k=2$ and $N_1=(1,0,0)$ and $N_2=(0,1,0)$. in this case $image(V)=(0,0,1)$2012-12-12
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    Your example seems to suggest that by "image of $v$ over $I$" you mean the orthogonal projection of $v$ onto $I$? By the way, mathematical variables are case-sensitive. It seems that you wanted $v$ and $V$ to refer to the same variable, but they don't. Also, usually vectors would be denoted by lower-case variables.2012-12-12
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    @joriki fixed case problem. besides that was only an example, I want to solve the problem in general case, meaning $n_i$ can be any vector in $R^n$ space. here is another example: $n=3, v=(0,0,1), k=2, n_1=(1,0,-1), n_2=(0,1,-1) => image(v) = (0.57,0.57,0.57)$2012-12-12
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    You didn't answer my question, but from this new example it seems that you *don't* mean the orthogonal projection of $v$ onto the intersection of the hyperplanes (which would be $(1/3,1/3,1/3)$ in that case). That brings us back to @copper.hat's question: What *do* you mean by the image of $v$? (Please give a definition, not another example.)2012-12-12
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    @joriki it seem I was looking for the orthogonal projection after all, but I was not familiar with the term.2012-12-12
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    @Gajoo: It would make things so much less cumbersome if you'd reply to comments comprehensively, rather than one aspect at a time, leaving other aspects unresolved. Do you disagree with my assertion that the orthogonal projection in your second example would have been $(1/3,1/3,1/3)$? If so, why? If not, how is that compatible with your statement that you're looking for the orthogonal projection after all?2012-12-12
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    @joriki I thought my mistake (computing projection vector) was obvious . and it makes your results the correct one.2012-12-12
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    If you are describing hyperplanes by their normal vectors and you want then to consider vectors that lie within those hyperplanes, you must actually want those vectors that are orthogonal to the original (normal) vectors. Perhaps your question should be worded (?) as: component of a vector $v$ that is orthogonal to a set of vectors $n_i$ Such a question could easily be answered here.2012-12-12

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