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So, how can we find it out in this case?

yes, dotproduct will determine if a vector is parralel to another.

I know my logic is flaw at somewhere, but don't know how to fix yet

k-ko that is

abcd
1100
-----
a+b = 0
abcd
-1010
------
-a+c= 0
a = a
b = -a
c = a
d = d

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but dotproduct is not cross product ... so i think your attempts are misguided

The goal of cross product is to find out a vector perpendicular to the given vectors, right?

Hence if we can't find it out by cross while dot can help us, why not?

that is an application of it yes, but i am not sure if that is the only reason for it.

Do you know something about flat ?

I read a paper but I don't get what it is.

'flat' doest ring any bells ...

ok, thanks for the help. :)

there are going to be an infinite number of vectors in R^4 that are perpendicular to a R^2 object.

a b c d e
-3,1,1,0,1
-----------
-3a+b+c+e = 0
a b c d e
-1,1,0,1,0
------------
-a+b+d = 0

or (a,b,0,0,0) or .... whatever is simplest for you to deduce

I will. Thank you so much. :)

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