Assume $a$, $b$ and $c$ are 3-dimensional vectors. When $a$, $b$ and $c$ are linearly dependent to each other, does the vector which is perpendicular to $a$, $b$ and $c$ exist all the time?
I learned that if it exists, then $a$, $b$ and $c$ are linearly dependent to each other, but I can't decide if the converse proposition would be true.
If $a$, $b$, and $c$ are linearly dependent, then one of them is a linear combination of the other two. Suppose, say, that $c$ is a linear combination of $a$ and $b$. Take a vector $d$ perpendicular to both $a$ and $b$. Then $d$ is also perpendicular to $c$.