H2 Math Vectors: Plane Equations, Angles and Shortest Distance
With Mr Gan5 min 20 secPublished
Watch Mr Gan solve 2018 H2 Math Paper 2 Question 3: find a plane equation, calculate an angle between planes and find the shortest distance from a point to a plane.
What you will learn
- Find a normal vector using two non-parallel vectors in a plane.
- Use the normals to calculate the acute angle between two planes.
- Project onto a unit normal to find the shortest distance.
Lesson notes
A written summary of the method, with mathematical notation clarified for revision.
Build the plane equation and check it
In a parallelogram, opposite sides have equal displacement vectors. Use that relationship to obtain the missing vertex before forming vectors in the required plane.
The cross product of two non-parallel vectors in the plane gives a normal n. With a known point a, the plane equation is r · n = a · n. A non-zero multiple of the normal describes the same plane.
Substitute the other known points into your plane equation. Mr Gan uses this check in the lesson to catch a sign error before continuing.
Use normals for angles and distances
For the acute angle θ between planes with normals n₁ and n₂, use cos θ = |n₁ · n₂| / (|n₁||n₂|). The absolute value selects the acute angle between the planes.
For a point P and any point A on the plane, the perpendicular distance is |AP · n| / |n|. If n is already a unit vector, the distance is simply |AP · n|.
In the worked question, find the midpoint of AD first, then project its displacement from a point on plane BCE onto the normal.
Try it yourself
Before replaying the angle calculation, identify which two normals are needed. Explain why the angle between two arbitrary lines in the faces would not answer the question.
Check your reasoning
Use a normal to the base plane and a normal to plane BCE. The normals determine the planes' orientations; arbitrary lines within the faces do not.