Angles, intersections and distances in space
Find the angle between a line and a plane, where a line meets a plane, the distance of a point from a plane and the angle between two planes, using scalar products and equations.
- UK: Year 13
- France: grade 12
- Germany: grade 12
- Iran: grade 12
- Syria: grade 12
- Russia: grade 11
Guide
This studio is not in Afghanistan's Year 11 book; it follows Syria's and Russia's books and UK Further Maths. The angle θ between a line (direction vector d) and a plane (normal vector n) comes from its sine: sinθ is the size of the scalar product d·n divided by the product of the two lengths. The angle between two planes is the acute angle between their normals.
To find where the line r = a + td meets a plane, put the coordinates of a point of the line into the plane's equation and solve for t. The distance of a point P from the plane Ax + By + Cz = D is the size of (the left side at P minus D) divided by the length of the normal.
Common mistakes: using the cosine instead of the sine for the angle with a plane, forgetting to divide, and taking the obtuse angle instead of the acute one between two planes.