3D vectors: lines and planes

<p>Acute angle between two planes: \(\cos \theta =\frac{\left| {{\mathbf{n}}_{1}}\cdot {{\mathbf{n}}_{2}} \right|}{\left| {{\mathbf{n}}_{1}} \right|\left| {{\mathbf{n}}_{2}} \right|}\)</p>
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For planes with normals \(\mathbf n_1,\mathbf n_2\), parallel normals give parallel or coincident planes; nonparallel normals give an intersection line.

\[\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|},\qquad 0\leq\theta\leq\frac\pi2.\]

Acute angle

\[\mathbf d=\mathbf n_1\times\mathbf n_2,\qquad \mathbf r=\mathbf a+\lambda\mathbf d,\]

Intersection line

Find any one common point \(A\) by solving the two plane equations (choosing a convenient free coordinate), then use the cross product direction.

Plane-Plane Relationships

Plane-Plane Relationships

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