Introduction to Three Dimensional Geometry

In Class 10 you worked on a flat paper with x and y. That is 2D. But the world is not flat. Your room has length, width and height. To say where a bulb is in your room, you need three numbers: how far from one wall, how far from second wall, and how high from floor.

That is why we add z-axis. x is left-right, y is front-back, z is up-down. All three meet at origin O(0,0,0). This is how we fix any point in space. Without 3D, you cannot tell a drone where to fly, you cannot build a building.

Take three sticks and join them at one end such that each is at 90 degrees to other two. Those are x,y,z axes. Pair of axes makes a plane: xy-plane is floor, yz-plane is one wall, zx-plane is other wall.

These three planes divide space into 8 parts called octants. Like 4 quadrants in 2D, but double. Octant I has all x,y,z positive. In your room corner, floor is xy, left wall yz, front wall zx. Sign of x,y,z tells you which octant you are in.

XY-plane is where z=0. All points on floor have no height. YZ-plane is where x=0, like left wall. ZX-plane is where y=0, like front wall. If a point lies on xy-plane, its z is zero. Example (3,4,0) lies on xy-plane.

If a point is on x-axis, then y=0 and z=0. On y-axis, x=0,z=0. On z-axis, x=0,y=0. This helps to quickly know where a point is. (0,0,5) is on z-axis 5 units up.

To get coordinates of a point P, drop perpendiculars to three planes. Distance from yz-plane is x-coordinate, from zx-plane is y-coordinate, from xy-plane is z-coordinate. So P(x,y,z).

Example: In your room, corner is origin. Walk 3 steps along length (x=3), 2 steps breadth (y=2), then height of bulb 4 ft (z=4) → point (3,2,4). Signs tell direction. Negative x means behind origin.

If you are given (2,-3,5), x=2 forward, y=-3 backward, z=5 up. So you can picture exactly where it is.

In 2D, distance between (x1,y1) and (x2,y2) is √((x2-x1)²+(y2-y1)²). In 3D, we just add z also. So distance between P(x1,y1,z1) and Q(x2,y2,z2) is √((x2-x1)²+(y2-y1)²+(z2-z1)²). It comes from applying Pythagoras two times.

Example: P(1,2,3) Q(4,6,8): difference 3,4,5 → distance √(9+16+25)=√50=5√2. If one point is origin, distance is √(x²+y²+z²) which is same as length of vector.

This is used to check if points form equilateral triangle in space, or to find radius if centre and point on sphere given.

You have two points A and B in space. You want a point P that divides AB in ratio m:n. Like you want a point 2:1 closer to B. Formula is similar to 2D but with z also.

Internal division: P = ( (m·x2+n·x1)/(m+n), (m·y2+n·y1)/(m+n), (m·z2+n·z1)/(m+n) ). The m goes with second point, n with first. If m=n, you get midpoint = average of A and B.

External division: P divides AB outside, then use minus: (m·x2-n·x1)/(m-n) etc. Example A(1,2,3) B(5,6,7) ratio 2:1 internal → x=(2*5+1*1)/3=11/3≈3.67, y=14/3≈4.67, z=17/3≈5.67.

Midpoint is simple average of two points. Centroid of triangle with vertices (x1,y1,z1),(x2,y2,z2),(x3,y3,z3) is average of three: ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). Similarly for tetrahedron with 4 points, divide sum by 4.

Example triangle A(1,2,3) B(2,3,4) C(3,4,5): centroid = ( (1+2+3)/3=2, (2+3+4)/3=3, (3+4+5)/3=4 ) → (2,3,4). This is centre of mass.

These formulas save time in many problems where you need centre of triangle or to prove something is isosceles in space.

3D geometry is everywhere. Your phone GPS gives latitude, longitude and altitude — that is x,y,z. Pilot needs x,y,z to land. Architect uses 3D coordinates to make building plan on computer.

Video games: Every character position is (x,y,z). When you move joystick, x,y,z changes. Construction: Crane operator knows exact 3D point to drop material. Robot arm in factory moves to (x,y,z) to pick object.

Medical: MRI scan shows point inside body in 3D. Chemist: Position of atoms in molecule is given in 3D. Even cricket: Hawk-eye tracks ball in 3D to decide LBW.