Three Dimensional Geometry

Look around — everything you see has length, breadth and height. That is 3D. A building is not just a rectangle, it is a box with coordinates (x, y, z). Architect uses (0,0,0) as corner and then every pillar is at some (x1,y1,z1). This is 3D coordinate system.

GPS is 3D Geometry live: Your phone location is not just latitude-longitude, it is (x, y, z) = latitude, longitude, altitude. Satellite in space knows your distance using formula distance = √[(x2-x1)²+(y2-y1)²+(z2-z1)²]. With 4 satellites it solves for your exact (x,y,z). That is trilateration - 3D geometry.

Drone path is a line: Drone flying from point A to point B follows line r = a + λb. a is starting point, b is direction. If drone goes from (0,0,10) in direction (1,2,0), its path is r = (0,0,10) + λ(1,2,0). For every λ, you get new position.

Aeroplane and runway are skew lines: Runway on ground and flight path in air - they never meet and are not parallel. Distance between them is shortest distance between skew lines. Pilots need to keep safe separation = |(a2-a1)·(b1×b2)| / |b1×b2|.

Video games, satellites: Game world is made of 3D coordinates. Satellite orbit plane is equation n·(r-a)=0. Wall meeting floor - angle between planes tells you if wall is 90° vertical.

Suppose a line makes angles α with X-axis, β with Y-axis, γ with Z-axis. Then we define direction cosines:

l = cos α, m = cos β, n = cos γ. These three numbers tell exact direction of line. Important identity: l² + m² + n² = 1 always. So if you know two, you can find third.

Example: If line makes 60° with X, 60° with Y, then l=0.5,m=0.5, l²+m²=0.5, so n²=0.5 => n=1/√2 => γ=45°. So line equally inclined to X,Y is at 45° to Z.

Direction Ratios (DR): Any numbers a,b,c proportional to l,m,n. Means if l,m,n are messy decimals, we use simple multiples. If DR are a,b,c, then magnitude R = √(a²+b²+c²) and l = a/R, m = b/R, n = c/R.

Example: DR = 2,3,6 => R = √(4+9+36)=7 => DC = 2/7,3/7,6/7. Check: (2/7)²+(3/7)²+(6/7)²=49/49=1. Perfect.

Two points P(x1,y1,z1) Q(x2,y2,z2): DR of PQ = x2-x1, y2-y1, z2-z1.

How to write a line in 3D? You need one point it passes through and its direction.

Vector form: If line passes through point A with position vector a and direction vector b, then any point P on line with position vector r is: r = a + λ b. λ is parameter. When λ=0, you are at A. When λ=1, you move by b. When λ changes, you slide along line.

Cartesian form: Let A = (x1,y1,z1), direction b = (a,b,c) [DR]. Let P=(x,y,z). Then (x-x1)/a = (y-y1)/b = (z-z1)/c = λ.

Example: Line through (1,2,3) direction (2,-1,4). Vector: r = (1,2,3)+λ(2,-1,4). Cartesian: (x-1)/2 = (y-2)/-1 = (z-3)/4.

Line through two points: A(x1,y1,z1) B(x2,y2,z2): Vector r = a + λ(b-a). Cartesian: (x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1).

If a DR is zero, e.g. b=0, then line is parallel to XZ plane, equation becomes (x-x1)/a = (z-z1)/c and y=y1.

Two lines in space have directions b1 and b2. Angle θ between them is angle between their direction vectors.

Formula: cos θ = |b1·b2| / (|b1||b2|). Dot product b1·b2 = a1a2+b1b2+c1c2. So cosθ = |a1a2+b1b2+c1c2| / [√(a1²+b1²+c1²) √(a2²+b2²+c2²)]. We take modulus to get acute angle (0° to 90°).

Parallel: If b1 is multiple of b2 => a1/a2 = b1/b2 = c1/c2 => θ=0°.

Perpendicular: If b1·b2=0 => a1a2+b1b2+c1c2=0 => θ=90°.

Example: b1=(1,2,3), b2=(4,0,1). Dot=1*4+2*0+3*1=7. |b1|=√14, |b2|=√17. cosθ=7/√(238)=0.454 => θ≈63°.

In vector form lines r=a1+λb1 and r=a2+μb2, only b1,b2 matter for angle, not a1,a2.

In 2D, two lines either intersect or are parallel. In 3D, there is third possibility: skew lines - they are not parallel and never meet, like opposite edges of a room. There is always a shortest distance between them which is perpendicular to both.

General lines: L1: r = a1 + λ b1, L2: r = a2 + μ b2.

Formula: SD = |(a2-a1)·(b1×b2)| / |b1×b2|. Numerator is triple product, denominator is magnitude of cross product.

What does it mean? (b1×b2) is vector perpendicular to both lines. (a2-a1) is vector joining a point on L1 to point on L2. Dot product projects that joining vector onto perpendicular direction, then dividing gives perpendicular length.

Intersecting condition: If lines intersect, SD=0 => (a2-a1)·(b1×b2)=0.

Parallel lines: b1×b2=0, so formula fails. Use: Distance = |(a2-a1)×b1| / |b1|. This is area of parallelogram / base.

Example: Real life - distance between drone cable and building edge, runway and flight path.

Plane is flat surface extending infinite - like floor, wall, table top. To define plane you need one point and normal vector (perpendicular to plane).

Vector form: Plane through point A(position vector a) with normal vector n: (r-a)·n = 0 or r·n = d where d=a·n. Why? Vector (r-a) lies in plane, n is perpendicular, so dot=0.

Cartesian: If A=(x1,y1,z1), n=(a,b,c), r=(x,y,z): a(x-x1)+b(y-y1)+c(z-z1)=0 => ax+by+cz+d=0. Here (a,b,c) are DR of normal.

Intercept form: If plane cuts axes at (p,0,0),(0,q,0),(0,0,r): x/p + y/q + z/r =1. Useful for quick sketch.

Plane through 3 points: A,B,C non-collinear: equation |x-x1 y-y1 z-z1; x2-x1 y2-y1 z2-z1; x3-x1 y3-y1 z3-z1|=0 determinant =0.

Normal form: If distance from origin is d and direction cosines of normal are l,m,n, then plane: lx+my+nz=d.

Example: Floor is plane z=0 with normal (0,0,1). Wall is plane x=0 with normal (1,0,0).

Angle between two planes: Angle between planes = angle between their normals n1,n2. So cosθ = |n1·n2| / (|n1||n2|) = |a1a2+b1b2+c1c2|/(√(a1²+b1²+c1²)√(a2²+b2²+c2²)). If n1·n2=0 => planes perpendicular. If n1 proportional to n2 => parallel.

Angle between line and plane: Line direction b, plane normal n. If line makes φ with plane, then it makes 90°-φ with normal. So sinφ = |b·n| / (|b||n|). If b·n=0 => line parallel to plane. If b proportional to n => line perpendicular to plane.

Distance from point to plane: Point P(x1,y1,z1), plane ax+by+cz+d=0: Distance = |ax1+by1+cz1+d| / √(a²+b²+c²). This is length of perpendicular from point to plane.

Distance between parallel planes: Planes ax+by+cz+d1=0 and ax+by+cz+d2=0: Distance = |d1-d2| / √(a²+b²+c²). Take any point on one plane and find its distance to other.

Real life: Wall perpendicular to floor => 90°. Drone height above ground plane = point to plane distance.

Final quick map of entire chapter for exam morning.

DC/DR: l=cosα,m=cosβ,n=cosγ, l²+m²+n²=1, l=a/√(a²+b²+c²).

Line: r=a+λb, (x-x1)/a=(y-y1)/b=(z-z1)/c.

Angle lines: cosθ=|b1·b2|/(|b1||b2|).

Skew SD: |(a2-a1)·(b1×b2)|/|b1×b2|, intersect if 0, parallel |(a2-a1)×b|/|b|.

Plane: (r-a)·n=0, r·n=d, ax+by+cz+d=0, x/p+y/q+z/r=1.

Angle plane-plane: cosθ=|n1·n2|/(|n1||n2|), line-plane sinθ=|b·n|/(|b||n|).

Distance: Point to plane |ax1+by1+cz1+d|/√(...), parallel planes |d1-d2|/√(...).