Concept 05 / 06Three Dimensional Geometry
Equation of a Plane: Normal, Intercept, Three-Point Forms and Angles Between Planes
The plane through a point with a given normal, the general equation ax + by + cz + d = 0, normal form r.n = p, intercept form, the plane through three points, the angle between two planes, parallel and perpendicular planes, and the family of planes through a line.
A plane is determined by a point on it and a normal vector (a vector perpendicular to every line in the plane).
Point-normal form. The plane through with normal consists of the points for which lies in the plane, i.e. is perpendicular to :
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In Cartesian form, with and the point : .
General form: . The coefficients of are the direction ratios of the normal. This is the most important reading skill in the topic.
Normal form: , i.e. with the direction cosines of the normal and the perpendicular distance from the origin. From the general form, divide by , choosing the sign that makes the right side positive.
Intercept form: , cutting the axes at , and , as in the figure. The area of the triangle cut off in the first octant and the volume of the tetrahedron with the origin () are common questions.
Plane through three points . The normal is . Equivalently, , i.e.
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Four points are coplanar iff the fourth satisfies this equation.
Special planes. The -plane is . A plane parallel to the -axis has no term: . A plane through the origin has .
Angle between two planes is the angle between their normals: . Perpendicular planes: . Parallel planes: . A plane parallel to is .
Family of planes through a line. If a line is the intersection of and , then every plane through that line (except itself) is . Impose one more condition (passing through a point, or perpendicular to a given plane) to find . This avoids computing the line explicitly.
Plane through a point containing a given line, or containing two intersecting lines: the normal is the cross product of two direction vectors lying in the plane.
Key Takeaways:
- ; .
- In , is the normal.
- Normal form ; intercept form .
- Three points: normal , or the determinant equation.
- Angle between planes angle between normals; perpendicular iff .
- Planes through the line : .
- Pitfall: treating a direction vector lying in the plane as its normal.
Brain-Teaser 1: Find the plane through , and .
Solution: , . Normal: . Plane: , i.e. . Check with : . Correct.
Brain-Teaser 2: Find the plane through the line of intersection of and that passes through .
Solution: The family: . At : , so . Multiplying by : , i.e. . Check: . The family method needed no explicit direction for the line.
Brain-Teaser 3: A plane meets the axes at so that the centroid of triangle is . Find the plane and the volume of the tetrahedron .
Solution: With intercepts , the centroid is , so , , . The plane is , i.e. . The volume is . The intercept form turns the centroid condition into three one-line equations.
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