Formula Sheet — Introduction to 3D Geometry

Card 1: The coordinate system

Fact Statement
Axes three mutually perpendicular lines through the origin O(0,0,0)O(0,0,0)
Coordinate planes XYXY (z=0z = 0), YZYZ (x=0x = 0), ZXZX (y=0y = 0)
Octants the 3 planes split space into 8 octants, named by the signs of (x,y,z)(x, y, z)
Meaning of the triplet x,y,zx, y, z = perpendicular distances from the YZYZ-, ZXZX-, XYXY-planes
On an axis xx-axis: (x,0,0)(x, 0, 0); yy-axis: (0,y,0)(0, y, 0); zz-axis: (0,0,z)(0, 0, z)
In a plane XYXY: (x,y,0)(x, y, 0); YZYZ: (0,y,z)(0, y, z); ZXZX: (x,0,z)(x, 0, z)

Card 2: Distances

Quantity Formula
Between two points PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}
From the origin x2+y2+z2\sqrt{x^2 + y^2 + z^2}
From the XYXY-plane ∣z∣\lvert z\rvert (one coordinate)
From the zz-axis x2+y2\sqrt{x^2 + y^2} (two coordinates)
Collinearity PQ+QR=PRPQ + QR = PR
Right triangle largest square == sum of the other two
Rectangle test parallelogram (equal opposite sides) ++ equal diagonals

Card 3: Section formula (JEE Corner)

Tool Statement
Internal m:nm : n (mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)
External m:nm : n replace nn by −n-n
Midpoint average each coordinate
Centroid (x1+x2+x33,y1+y2+y33,z1+z2+z33)\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)
Fourth vertex of ABCDABCD D=A+C−BD = A + C - B
XYXY-plane cuts ABAB in −z1:z2-z_1 : z_2 (use xx's for YZYZ, yy's for ZXZX)
Sphere x2+y2+z2+2ux+2vy+2wz+d=0x^2+y^2+z^2+2ux+2vy+2wz+d = 0: centre (−u,−v,−w)(-u,-v,-w), radius u2+v2+w2−d\sqrt{u^2+v^2+w^2-d}

Last-Minute Mistake Checklist

  1. Distance from a plane vs from an axis: from the XYXY-plane it is ∣z∣|z|; from the zz-axis it is x2+y2\sqrt{x^2 + y^2}. One coordinate vs two — never interchange them.
  2. Octant naming: memorise the sign table as "I-IV have z>0z > 0 cycling the 2D quadrants; V-VIII repeat them with z<0z < 0" — reconstructing beats recalling eight rows.
  3. Compare squared distances: 18\sqrt{18} and 323\sqrt{2} are the same number; squares make equalities visible and save surd slips.
  4. Right-triangle check: test the largest squared side against the sum of the other two — testing a random pairing produces false negatives — only the largest square can be the hypotenuse's.
  5. Rectangle needs more than a parallelogram: equal opposite sides is not enough; verify equal diagonals too.
  6. Reflections: in a coordinate plane, one coordinate flips; in an axis, two flip; in the origin, all three flip.
  7. Section formula weights: the ratio m:nm : n puts mm on the far point BB — writing mx1+nx2m+n\frac{mx_1 + nx_2}{m+n} silently swaps the ratio.
  8. External division: replace nn by −n-n throughout, including the denominator m−nm - n; a common slip is negating only the numerator.
  9. Plane-division sign: ratio −z1:z2-z_1 : z_2 positive means an internal cut (endpoints on opposite sides); negative means external. State which.
  10. Locus questions: write P(x,y,z)P(x, y, z) first, translate the sentence into distances, square once, simplify fully — an unsimplified locus loses the final mark.

How this chapter flows on: the next chapter (Limits and Derivatives) changes gears into calculus; these coordinates return in Class 12 as the stage for direction cosines, straight lines and planes in space — with the distance and section formulas assumed known.