I am thinking on a question of how to find a linear homogeneous equation such that $(1,1,1,1)$,$(1,-1,-1,1)$ and $(2,3,3,2)$ are solutions of the equation.So far I think I am after a hyperplane in four dimensional space which contains these given vectors,but I have no Idea about how to find the equation.Can someone please help me a bit.I think I am looking for the hyperplane spanned by these vectors.Please somebody provide me with an answer on this question,so that I can obtain a linear homogeneous equation with prescribed solution.
2026-04-13 04:41:41.1776055301
Find a homogeneous linear equation in $4$ variables with $3$ given vectors as solution.
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The hyperplane spanned by those three vectors has a normal vector $\vec{n}$. Each point in this plane has a position vector $\vec{v}$ which obeys the equation
$$\vec{n} \cdot \vec{v} = 0.$$
This is the linear equation you're after. I determined on sight that $(1,0,0,-1)$ is a normal vector.