Line and Plane

We just learned how to describe lines in 3D. Now let's add planes into the mix. Planes are flat surfaces that extend infinitely — think of a tabletop, a wall, or the surface of water. And when lines and planes interact, we get interesting questions: where does a line pierce through a plane? What's the angle between a line and a plane? How far is a point from a plane? What about the line where two planes intersect? This chapter answers all of these questions and more. It's the geometry of how things meet in 3D space, and it's used in everything from architectural design to computer graphics to calculating satellite orbits. We start with the equation of a plane. The simplest form is the vector equation: r·n = d, where n is a vector perpendicular to the plane (the normal vector) and d is a constant. In Cartesian form, this becomes ax + by + cz = d, where the coefficients (a, b, c) are the components of the normal vector. If d = 0, the plane passes through the origin. There's also the normal form: r·n̂ = p, where n̂ is a unit normal and p is the perpendicular distance from the origin to the plane. And the intercept form: x/a + y/b + z/c = 1, where a, b, c are the x, y, z intercepts respectively. Each form is useful in different situations. Next, we learn how to find the equation of a plane under different conditions. A unique plane is determined by: (i) a point and a normal vector, (ii) three non-collinear points, (iii) a line and a point not on the line, or (iv) two intersecting lines. For three points, find two vectors in the plane, take their cross product to get the normal, then plug a point into r·n = d. For a line and a point, take the direction of the line and the vector from a point on the line to the given point, cross them to get the normal. Then we tackle angles. The angle between two planes is simply the angle between their normal vectors: cos θ = |n₁·n₂| / (|n₁||n₂|). If n₁·n₂ = 0, the planes are perpendicular. If n₁ and n₂ are parallel, the planes are parallel. For the angle between a line and a plane, we take the complement of the angle between the line and the normal: sin φ = |b·n| / (|b||n|), where b is the line's direction and n is the plane's normal. The distance of a point from a plane has a clean formula: d = |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²). This is the perpendicular distance — plug the point into the plane equation, take absolute value, divide by the normal vector's magnitude. For two parallel planes, the distance between them is the difference of their d-values divided by the normal magnitude. We also cover the intersection of two planes. Two non-parallel planes always intersect in a straight line. The direction of this line is n₁ × n₂ (perpendicular to both normals). To find a point on the line, set one variable (like z) to 0 and solve the two plane equations. Once you have a point and a direction, you have the line. Finally, we look at coplanarity of lines — whether two lines lie in the same plane. Two lines are coplanar if the scalar triple product (a₂ − a₁)·(b₁ × b₂) = 0, where a₁, a₂ are points on the lines and b₁, b₂ are their directions. If this is zero, they either intersect or are parallel — both are coplanar. If non-zero, they're skew (covered in the 3D geometry chapter). By the end of this chapter, you'll be able to write plane equations from any given data, find intersections and angles involving planes, and solve problems that combine lines and planes in 3D space.