Can 3 planes fail to intersect even if none of the planes are parallel? I am working out of textbook on Linear Algebra. One of the questions states categorically:

"Three planes can fail to have an intersection point, even if no planes are parallel."

This seems counter intuitive.  Is this true? If so how?
 A: Any two of them must intersect, if no two are parallel, but there need not be a point that all three of them have in common. The intersection of the first two is a line $\ell$. Let $\ell'$ be a line different from but parallel to $\ell$. There are infinitely many planes through $\ell'$, but only one of them intersects $\ell$, and only two of them are parallel to one of the first two planes. Thus, there are infinitely many of them that fail to intersect $\ell$ and are not parallel to either of the first two planes.
A: Assuming you are working in $\Bbb R^3$, if the planes are not parallel, each pair will intersect in a line.  There is nothing to make these three lines intersect in a point.  The text is taking an intersection of three planes to be a point that is common to all of them.
A: $x=0$, $y=0$, $x+y=1$. (Each line of intersection of two of the planes is parallel to the third.)
A: Since this question comes from a linear algebra textbook, let's consider it from a linear algebraic perspective. Every plane (in $\mathbb{R}^3$, I suppose) can be represented by the equation $\mathbf{u}^T\mathbf{x}=d$, where $\mathbf{u}$ denotes the unit normal (pointing from the original) to the plane and $d$ denotes the distance from the origin to the plane. So, three planes $\mathbf{u}_i^T\mathbf{x}=d_i\ (i=1,2,3)$ intersect if and only if
$$
\underbrace{\pmatrix{\mathbf{u}_1^T\\ \mathbf{u}_2^T\\ \mathbf{u}_3^T}}_{\mathbf{A}}\ \mathbf{x}
=\underbrace{\pmatrix{d_1\\ d_2\\ d_3}}_{\mathbf{b}}.
$$
Therefore, the question is essentially asking if the matrix equation $\mathbf{A}\mathbf{x}=\mathbf{b}$, where $\mathbf{A}$ is $3\times3$ and $\mathbf{b}\in\mathbb{R}^3$, always has a solution $\mathbf{x}$. I think you should have learnt the answer in the lectures.
A: Intution: two planes intersect in a line.  For the third plane not to intersect that line at a point, it must be parallel to the line.  Now the picture: imagine walking down an infinite triangular corridor, always the same size.  That is what the three planes produce.
