# Let$(a,b,c)$ be a nonzero vector in the row space of a $3\times 3$ matrix $B$. Show that the nullspace of $B$ is a subset of the plane $ax+by+cz=0$

Let $(a$ $b$ $c)$ be a nonzero vector that belongs to the row space of a $3\times 3$ matrix $B$. Show that the nullspace of $B$ is a subset of the plane $ax + by + cz = 0.$

My thoughts so far: Let $\alpha=(x$ $y$ $z)$ be a nonzero vector in the nullspace of $A.$ If I can show $(a$ $b$ $c)\alpha=0$ then the proof is completed. However I don't know how to utilize this piece of information:

$(a$ $b$ $c)$ be a nonzero vector that belongs to the row space of a $3\times 3$ matrix $B$

Desperately need a hint on this, thank you!

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## 1 Answer

Consider that the set of vectors in the plane are orthogonal to the normal vector. What can you say about the orthogonality between the rows of a matrix and a vector in it's nullspace?

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the rowspace of a matrix is the orthogonal complement to its nullspace? – drawar Nov 16 '12 at 17:20
Yes. Does that help you answer your question? – EuYu Nov 16 '12 at 18:55
That definitely helps, thanks! – drawar Nov 17 '12 at 6:15