⁦Angles, intersections and distances in space⁩

⁦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.