Relative positions and distances in space

What is it about?

Lines are written in parametric form $g\colon \vec x = \vec a + t \cdot \vec u$, planes in coordinate form $E\colon n_1x_1 + n_2x_2 + n_3x_3 = c$ with the normal vector $\vec n$.

Two lines: if the directions are parallel, the lines are identical or parallel; otherwise they either intersect (the system $\vec a + t\vec u = \vec b + s\vec v$ has a solution) or are skew. The angle between lines satisfies $\cos\varphi = \frac{|\vec u \cdot \vec v|}{|\vec u||\vec v|}$; skew lines have the distance $d = \frac{|(\vec b - \vec a)\cdot(\vec u \times \vec v)|}{|\vec u \times \vec v|}$.

Line and plane: if $\vec n \cdot \vec u = 0$ the line lies in $E$ or is parallel to it; otherwise there is one point of intersection, and $\sin\varphi = \frac{|\vec n \cdot \vec u|}{|\vec n||\vec u|}$.

Two planes are parallel (or identical) when their normals are parallel; otherwise they meet in a line with direction $\vec n \times \vec m$.

Distance point – plane (Hesse normal form): $d(P; E) = \frac{|n_1p_1 + n_2p_2 + n_3p_3 - c|}{|\vec n|}$. The foot $F$ lies on the perpendicular $\vec x = \vec p + t\vec n$; going the same distance again gives the mirror point $P'$.

Try it

1. Play the animation for the skew lines and watch the path light up in the decision tree. Then use “View along g”. Load example
2. Drag the tip of $\vec v$ until $h$ meets $g$.
3. Nearly parallel lines meet far away at a very small angle. Small angle
4. View two intersecting planes along their line of intersection: the angle appears in true size. Line of intersection
5. Drop the perpendicular, switch on the mirror point and drag $P$ into the plane. Distance

Tasks

Task 1: Intersecting lines

Find the point and angle of intersection of $g\colon \vec x = (1, 0, 0) + t(1, 1, 0)$ and $h\colon \vec x = (0, 1, 0) + s(1, -1, 0)$. Load task

Task 2: Point of intersection with a plane

Intersect $g\colon \vec x = (0, 0, 5) + t(1, 1, -1)$ with $E\colon x_1 + x_2 + x_3 = 2$. Load task

Task 3: Distance point – plane

Find $d(P; E)$, the foot $F$ and the mirror point for $P(1, 2, 3)$ and $E\colon 2x_1 - x_2 + 2x_3 = 3$. Load task

Task 4: Parallel planes

Find the distance between $x_1 + x_2 + x_3 = 1$ and $x_1 + x_2 + x_3 = 4$. Load task

Ideas for the classroom

• Let students predict the path through the decision tree before playing the animation; each question is one step of the calculation.
• The edge-on views show why distances and angles appear in true size when looking parallel to a plane.
• Common stumbling blocks:
  • Skew lines look as if they cross in the oblique picture – rotating the view helps.
  • Using cosine instead of sine for the angle between a line and a plane.
  • Using the same parameter name for both lines.
  • Forgetting the absolute value or the division by $|\vec n|$ in the Hesse normal form.