🎯 Key Points
- Three mutually perpendicular axes meet at the origin; the three coordinate planes (XY, YZ, ZX) cut space into 8 octants
- A point is written P(x, y, z) — x is measured along OX, y along OY, z along OZ
- A point on the XY-plane has z = 0; on the YZ-plane x = 0; on the ZX-plane y = 0. On the x-axis both y = 0 and z = 0
- Distance PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] — the 2D formula with one extra term
- Distance of P(x, y, z) from the origin = √(x² + y² + z²)
- Internal section in ratio m:n → ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)); for external division replace n by −n
- Midpoint is the m:n = 1:1 case → ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
- Centroid of a triangle → ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3)
Coordinate Axes and Coordinate Planes
Take three mutually perpendicular lines through a fixed point O. These are the x-axis, y-axis and z-axis, and O is the origin. Taken in pairs they determine three planes:
- XY-plane — contains the x and y axes; every point on it has z = 0
- YZ-plane — contains the y and z axes; every point on it has x = 0
- ZX-plane — contains the z and x axes; every point on it has y = 0
These three planes divide space into eight regions called octants, distinguished by the sign pattern of (x, y, z). The first octant is where all three are positive.
Coordinates of a Point in Space
To locate P, drop perpendiculars from P onto the three axes. The signed lengths cut off are its coordinates (x, y, z). Some useful special cases:
- Origin → (0, 0, 0)
- Point on the x-axis → (x, 0, 0); on the y-axis → (0, y, 0); on the z-axis → (0, 0, z)
- Point in the XY-plane → (x, y, 0)
Distance Between Two Points
For P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
This follows from applying Pythagoras twice — once in the base plane, once vertically. Setting the second point to the origin gives OP = √(x² + y² + z²).
Section Formula
If R divides the join of P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m : n, then
R = ( (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n) )
For external division in ratio m : n, replace n with −n throughout, giving denominators of (m − n). The midpoint is simply the case m = n = 1.
💡 Advanced Edge
- To test whether three points are collinear in 3D, compute all three pairwise distances — the largest must equal the sum of the other two. (Slopes don't exist in 3D, so the 2D slope test doesn't transfer.)
- A ratio that comes out negative when you solve the section formula means the point divides the segment externally, not internally.
- To find where the line joining two points crosses a coordinate plane, set the relevant coordinate to 0 in the section formula and solve for the ratio — e.g. for the XY-plane set the z-coordinate to 0, giving k = −z₁/z₂.
Example 1 — Distance
Q: Find the distance between P(1, −3, 4) and Q(−4, 1, 2).
Step 1 — Differences: Δx = −4 − 1 = −5, Δy = 1 − (−3) = 4, Δz = 2 − 4 = −2.
Step 2 — Square and add: 25 + 16 + 4 = 45.
Answer: PQ = √45 = 3√5 units.
Example 2 — Section formula
Q: Find the point dividing the join of A(1, −2, 3) and B(3, 4, −5) internally in the ratio 1 : 3.
Step 1 — Here m = 1, n = 3, so m + n = 4.
Step 2 — x: (1·3 + 3·1)/4 = 6/4 = 3/2.
Step 3 — y: (1·4 + 3·(−2))/4 = (4 − 6)/4 = −1/2.
Step 4 — z: (1·(−5) + 3·3)/4 = (−5 + 9)/4 = 1.
Answer: (3/2, −1/2, 1).
Example 3 — Crossing a coordinate plane
Q: In what ratio does the YZ-plane divide the line joining A(−2, 4, 7) and B(3, −5, 8)?
Step 1 — On the YZ-plane the x-coordinate is 0.
Step 2 — Let the ratio be k : 1, so x = (3k − 2)/(k + 1) = 0.
Step 3 — Solve: 3k = 2, so k = 2/3.
Answer: the YZ-plane divides AB internally in the ratio 2 : 3.