Suppose there are $2$ vectors such that vector dot product of them i.e $\langle a,b,c \rangle \cdot \langle x,y,z \rangle = C$ i.e $C$ is a scalar quantity. Is there a way to find out another vector such that when we do dot product with $\langle x,y,z \rangle$ it also gives $C$?
Example is $\langle 6,0,0 \rangle \cdot \langle 2,3,7 \rangle = 12$ and also $\langle 1,1,1 \rangle \cdot \langle 2,3,7 \rangle = 12$. So how do we find all such vectors such that when multipled by $\langle 2,3,7 \rangle $ it gives result $12$.
Is there any concept that I can use for the same?
Yes, there is a name for this concept. If your space is three-dimensional, it's called a plane. One usually writes the collection of all such vectors $[x, y, z]$ as $$ 2x+3y+7z = 12 $$ but your dot product notation $$ [x, y, z]\cdot [2, 3, 7] = 12 $$ also works.
It is called a "plane" because the collection of all such vectors (if you're used to thinking about vectors as arrows, then the collection of all possible end points of all those vectors, if they start at the origin) makes an entirely flat surface, which is what a plane is. This surface is orthogonal / normal to the vector $[2, 3, 7]$, and $[2, 3, 7]$ is therefore called a normal vector to that plane.