Spatial vectors, planes, and lines

Idea

The hard part of a problem in space is usually not the formula. It is keeping track of two things at once: where a vector sits, and which way a plane faces. This page keeps one scene fixed, a vector $\mathbf{v}$ and the plane spanned by $\mathbf{a},\mathbf{b}$. All three readings act on those two objects. The question does not change.

First the position of $\mathbf{v}$ is spread into shadows on the three coordinate planes. Then the cross product packs the facing direction of the plane into a single normal vector $\mathbf{n}$. Last, the same dot-product reading decides how $\mathbf{v}$ is related to that plane. The full derivation of each tool, and the degenerate cases, stay on the matching single work.

Interaction

• Position: drag the components $v_x,v_y,v_z$ and watch one position in space spread into three flat shadows
• Direction: change Plane tilt and watch the plane spanned by $\mathbf{a},\mathbf{b}$ pack into a single $\hat{\mathbf{n}}$
• Relation: switch among Line crosses the plane, v parallel to the plane, and v lies in the plane, and compare how the readings $\hat{\mathbf{n}}\cdot \mathbf{v}$ and $\hat{\mathbf{n}}\cdot \mathbf{v}-h$ change
• Drag the figure to rotate the view and check that the three readings are the same scene. Horizontal view and Elevation do the same rotation

Suggested order: Position, then Direction, then Relation.

What to notice

• The first two readings answer different questions. One asks where $\mathbf{v}$ is. One asks which way the plane faces. Only together do they decide how the two are related.
• The facing direction of a plane can be packed into one normal vector. After that, every line-plane judgment needs only that arrow, not the original $\mathbf{a},\mathbf{b}$.
• Lying in the plane, parallel, and crossing are not three formulas. They come from one dot product: if $\hat{\mathbf{n}}\cdot \mathbf{v}\ne 0$, the line through $O$ along $\mathbf{v}$ crosses the plane; if $\hat{\mathbf{n}}\cdot \mathbf{v}=0$, the line is parallel to the plane at distance $|h|$, and it lies in the plane exactly when $h=0$. The other reading, $\hat{\mathbf{n}}\cdot \mathbf{v}-h$, is the signed distance from the tip of $\mathbf{v}$ to the plane.

Related

• Geometric significance of the cross product
• Spatial vector and three plane projections
• Where a line meets a plane
• Plane normal and distance from a point

Further reading

• Euclidean vector (Wikipedia)
• Euclidean plane (Wikipedia)
• Line–plane intersection (Wikipedia)