Three Dimensional Geometry
Easy Overview
You've done geometry in 2D your whole life. Now imagine adding a third dimension — depth. This chapter gives you the tools to work with points, lines, and planes in 3D space. It's how GPS works, how 3D movies are made, and how architects design buildings.
Coordinates in 3D Space
Every point in space is (x, y, z). X is left-right, Y is forward-backward, Z is up-down. The origin (0, 0, 0) is where all three axes meet. Distance between two points? √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Same as 2D but with z added. The section formula also works the same way — just with three coordinates.
Direction Cosines and Direction Ratios
Every line in 3D has direction cosines (l, m, n) — the cosines of the angles it makes with the X, Y, and Z axes. Property: l² + m² + n² = 1. Direction ratios are proportional numbers (a, b, c) — easier to find, but give the same direction. If a line has ratios 2, 3, 4, you can find its actual cosines by dividing by √(a²+b²+c²).
Equation of a Line in 3D
A line is defined by a point and a direction. Vector form: r = a + λb. a is a point on the line, b is the direction. In Cartesian: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c = λ. Each λ gives you a different point on the line. Think of it like a train on tracks — a is the station, b is the direction the tracks go.
Angle Between Two Lines
The angle between two lines is just the angle between their direction vectors. If direction cosines are (l₁, m₁, n₁) and (l₂, m₂, n₂): cos θ = l₁l₂ + m₁m₂ + n₁n₂. For direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂): cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / √((a₁²+b₁²+c₁²)(a₂²+b₂²+c₂²)). If cos θ = 0, they're perpendicular.
Distance of a Point from a Line
To find how far a point is from a line, drop a perpendicular from the point to the line. The formula uses cross product: distance = |(r₂ − r₁) × b| / |b|. r₁ is a point on the line, r₂ is your point, b is the line's direction. It's the area of a parallelogram divided by its base.
Key Points
- •Distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂): √((Δx)² + (Δy)² + (Δz)²)
- •Direction cosines l, m, n satisfy l² + m² + n² = 1
- •Direction ratios (a, b, c) are proportional to (l, m, n)
- •Line through (x₁,y₁,z₁) with ratios (a,b,c): (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
- •Angle between lines: cos θ = l₁l₂ + m₁m₂ + n₁n₂
- •Lines are perpendicular if l₁l₂ + m₁m₂ + n₁n₂ = 0
- •Skew lines never intersect and are not parallel
Practice Questions
- Find the direction cosines of the line passing through (2, 3, −1) and (4, 6, 3).
- Show that the lines (x−1)/2 = (y+1)/3 = (z−2)/4 and (x−3)/4 = (y−2)/6 = (z−1)/8 are parallel.
- Find the angle between the lines whose direction ratios are (1, 2, 2) and (2, 3, 6).
- Find the distance of point (1, 2, 3) from the line (x−1)/2 = (y−2)/3 = (z−3)/4.