Lines and planes in space

What is it about?

A line in space is given by a point $A$ and a direction. In parametric form: $g\colon \vec x = \vec a + t \cdot \vec u$, where $\vec a$ is the position vector of $A$ and $\vec u \ne \vec o$ the direction vector. Every value of $t$ gives exactly one point $X$. For $\vec a = (1, 2, 3)^T$, $\vec u = (2, -1, 1)^T$ and $t = 2$ you get $X(5, 0, 5)$.

• Trace points are where the line pierces a coordinate plane, e.g. $x_3 = 0$ gives $t = -3$ and $S_{12}(-5, 5, 0)$. If, say, $u_3 = 0$, the line is parallel to the $x_1x_2$ plane or lies in it, so there is no trace point in that plane (or the whole line lies in it).
• Point test: $P$ lies on $g$ if all three rows of $\vec p = \vec a + t \vec u$ give the same $t$.
• A plane needs two non-parallel spanning vectors: $E\colon \vec x = \vec a + r \vec u + s \vec v$.
• The normal vector $\vec n = \vec u \times \vec v$ is perpendicular to the plane. Normal form $\vec n \cdot (\vec x - \vec a) = 0$, coordinate form $n_1x_1 + n_2x_2 + n_3x_3 = d$ with $d = \vec n \cdot \vec a$.
• Example: $2x_1 + 3x_2 + 6x_3 = 12$ has the intercepts $S_1(6, 0, 0)$, $S_2(0, 4, 0)$, $S_3(0, 0, 2)$ – the trace triangle.

Try it

1. Drag $X$ along the line or move the slider $t$. Watch the arrow chain $\vec a + t \vec u$. Load example
2. Make $\vec u$ parallel to the $x_3$ axis. How many trace points are left? Parallel to the x₃ axis
3. Press “Convert” and follow the four steps to the trace triangle. Convert
4. Make $\vec v$ a multiple of $\vec u$. u and v parallel

Tasks

Task 1: Point test

Does $R(7, -1, 6)$ lie on $g\colon \vec x = (1, 2, 3)^T + t \cdot (2, -1, 1)^T$? Load task

Task 2: Coordinate form

Find a coordinate form and the intercepts of $E\colon \vec x = (1, 0, 0)^T + r \cdot (-1, 1, 0)^T + s \cdot (-1, 0, 1)^T$. Load task

Task 3: Trace triangle

Find the area of the trace triangle of $2x_1 + 3x_2 + 6x_3 = 12$.

Ideas for the classroom

• Use: upper secondary analytic geometry (grade 12/13), introducing parametric forms or practising intercepts.
• Parameter as time: $X$ travels through space with velocity $\vec u$ – this explains why $t$ may be negative.
• Common stumbling blocks:
  • Checking only two rows in the point test – all three must give the same $t$.
  • Losing a sign in the middle component of the cross product; the check $\vec n \cdot \vec u = 0$ reveals it.
  • Confusing trace points of a line (with coordinate planes) and of a plane (with the axes).