Three Dimensional Geometry

You've spent years doing geometry in two dimensions — plotting points on x and y axes, finding distances, slopes, and equations of lines. But the world we live in is three-dimensional. This chapter is your passport to working in 3D space. Every point now has three coordinates (x, y, z), and lines have direction in all three dimensions. This is the geometry behind 3D movies, GPS navigation, flight paths, architectural blueprints, and even the way your phone tracks its orientation in space. We start with the basics of the 3D coordinate system. There are three mutually perpendicular axes — X, Y, and Z — meeting at the origin. The space is divided into 8 octants (like quadrants in 2D, but now we have up and down too). The coordinates of a point P are (x, y, z) — the distances along the three axes. The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) — it's just the 2D distance formula with z added. The section formula extends naturally too: if R divides PQ internally in ratio m:n, then R = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n)). The midpoint is the special case with m = n. Next come direction cosines and direction ratios — these describe which way a line points. Direction cosines (l, m, n) are the cosines of the angles that the line makes with the X, Y, and Z axes. They satisfy the identity l² + m² + n² = 1. Direction ratios (a, b, c) are any three numbers proportional to (l, m, n). They're easier to find — you just take the differences in coordinates between two points on the line. To convert direction ratios to direction cosines: divide each by √(a² + b² + c²). Then we learn how to write the equation of a line in 3D space. In vector form, r = a + λb, where a is the position vector of a point on the line and b is the direction vector. In Cartesian (symmetric) form, (x − x₁)/a = (y − y₁)/b = (z − z₁)/c = λ. Each value of λ gives a different point on the line. The parametric form writes x = x₁ + aλ, y = y₁ + bλ, z = z₁ + cλ — useful for finding specific points. You'll learn to find the angle between two lines using their direction cosines (cos θ = l₁l₂ + m₁m₂ + n₁n₂) or direction ratios (cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / √((a₁²+b₁²+c₁²)(a₂²+b₂²+c₂²))). If the numerator is zero, the lines are perpendicular. If the direction ratios are proportional, they're parallel. We also cover conditions for a point to lie on a line, and how to find the distance of a point from a line using cross product. Finally, we tackle skew lines — lines that are neither parallel nor intersecting. They exist in different planes. The shortest distance between two skew lines is the length of the common perpendicular. The formula uses the scalar triple product: SD = |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|. This is the distance between the two lines at their closest approach. By the end of this chapter, you'll be comfortable navigating 3D space and describing lines in multiple equivalent forms. The skills here feed directly into the next chapter on line and plane, where we add planes to the mix.