Vectors: planes, normals and distances
Learn one idea at a time
Choose a unit. Each has worked examples and practice.
Describe a plane
Choose between parametric, normal and Cartesian forms.
Meet a line and a plane
Decide whether a line crosses, misses or lies in a plane, then find its angle.
Find perpendicular feet and distances
Use the normal to find a foot and measure a distance.
Compare two planes
Find their separation, common line or acute angle.
By the end, you can
- Describe planes using normals or two directions
- Find point-to-plane distance
- Relate angles and intersections to vector products
4 focused units · about 40 minutes for the full chapter. Allow extra time for practice.
The central idea
A plane can be written , where its nonzero normal vector is perpendicular to every direction in the plane. Equivalently, two independent in-plane directions give a parametric equation. The distance from P is .
Derive a parametric plane
Choose a fixed point A with position a and two nonparallel nonzero directions b,c parallel to the plane. A displacement in the plane is . Triangle addition gives , hence
This equation is not unique: another base point or independent pair of directions can describe the same plane. A drawn parallelogram is only a finite patch of an infinite plane and does not bound the real parameters. Parallel directions would not span a plane.
Worked example
The plane has normal with magnitude 3. The origin is distance from the plane. Its perpendicular foot is .
The normal vector is perpendicular to the plane; it is not a direction lying in the plane.
Check your understanding
A point is and a plane is . Find the perpendicular foot F, the distance from P to the plane and the reflection Q of P in the plane. Verify the plane, direction and midpoint conditions. Explain why writing the plane as must give the same results.
A normal is , with and . Put . The plane condition gives , so and . Indeed , and is parallel to the normal. The distance is 3.
The image is , and . With the reversed normal and constant, the parameter becomes and the displacement is still . Scaling both sides preserves the plane, its foot, distance and reflected point; only the chosen normal orientation changes.
Practice questions
40 examples and practice questions, in 7 sections.
More practice
Use geometry to choose a direction, normal or projection, then verify the resulting point in every required plane. Distinguish a line–plane angle from an angle between normals.
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