Line-Plane Relationships
Line-Plane Relationships
Let \(l:\mathbf r=\mathbf a+\lambda\mathbf d\) and \(\pi:\mathbf r\cdot\mathbf n=p\). The scalar \(\mathbf d\cdot\mathbf n\) controls the direction relationship.
Line contained in a plane
Line parallel to a plane
Line intersecting a plane
Line perpendicular to a plane
Angle between a line and a plane
Contained case
Parallel case
Acute intersection case
Perpendicular case
Projection onto a plane
| Condition | Conclusion |
|---|---|
| \(\mathbf d\cdot\mathbf n=0\) and \(\mathbf a\in\pi\) | Line lies in plane |
| \(\mathbf d\cdot\mathbf n=0\) and \(\mathbf a\notin\pi\) | Line is parallel to plane |
| \(\mathbf d\cdot\mathbf n\ne0\) | Line meets plane once |
| \(\mathbf d=\lambda\mathbf n\) | Line is perpendicular to plane |
\[\sin\alpha=\frac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|},\quad 0\leq\alpha\leq\frac\pi2;\qquad (\mathbf a+\lambda\mathbf d)\cdot\mathbf n=p.\]
\[\operatorname{proj}_{\pi}\mathbf v=\mathbf v-\frac{\mathbf v\cdot\mathbf n}{|\mathbf n|^2}\mathbf n.\]
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