Intersection of three planes

What you are looking at

The system here is $x + y + z = 3$, $x - y + 2z = 2$ and $2x + kz = m$. Row reduction turns the third equation into $(k - 3)z = m - 5$, and the determinant of the coefficients is $6 - 2k$. When $k \ne 3$ the determinant is non-zero and the three planes meet in exactly one point, drawn as a dot inside the box. When $k = 3$ there is no unique solution, and what happens next depends on $m$.

If $k = 3$ and $m = 5$, the last row reads $0z = 0$, so one variable is free and the three planes share a whole line, $\mathbf{r} = \begin{pmatrix} 2.5 \\ 0.5 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -1 \\ -2 \end{pmatrix}$. If $k = 3$ and $m \ne 5$ the system is inconsistent: no two planes are parallel, but each pair meets in a line and the three lines are parallel, forming a triangular prism. Dashed lines on the drawing show where each pair meets.

Try this

1. 1Start with $k = 1$ and $m = 4$: the planes meet at the single point $(1.75, 0.75, 0.5)$. Drag the drawing to rotate it and check that the dot lies on all three planes.
2. 2Move $k$ to 3 and leave $m = 4$. The label changes to no solution; rotate the box until you are looking along the three parallel dashed lines and the triangular prism appears.
3. 3Now set $m = 5$ as well. The third plane slides until it passes through the line where the first two meet, and the working gives the vector equation of the shared line.
4. 4Hide $\Pi_3$ with its button below the drawing. With two planes left, the panel gives their line of intersection, whose direction is $\mathbf{n}_1 \times \mathbf{n}_2 = \begin{pmatrix} 3 \\ -1 \\ -2 \end{pmatrix}$. Press Spin to turn the box on its own.

Mistakes students make

• Concluding that there is no solution as soon as the determinant is zero. A zero determinant only rules out a unique solution; the system can still be consistent, with infinitely many solutions along a line, as here when $k = 3$ and $m = 5$.
• Thinking that no solution means some of the planes are parallel. Here no two normals are parallel, yet for $k = 3$ and $m \ne 5$ the planes form a triangular prism with no point common to all three.
• Giving a line of intersection without a parameter. When a row reduces to $0 = 0$, let one variable equal a parameter such as $\lambda$ and write the others in terms of it, to reach the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$.

Where it is taught

• IB Maths AA HL 1.16 Systems of linear equations (up to three equations in three unknowns)
• IB Maths AA HL 3.18 Intersections of lines and planes