3D Geometry — NDA Mathematics

Formulas (3)

• Distance between two points · Distance between two points
  $$AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$
• Section formula — dividing a segment in a ratio · Internal division in ratio m : n
  $$\left( \frac{m x_2 + n x_1}{m+n},\ \frac{m y_2 + n y_1}{m+n},\ \frac{m z_2 + n z_1}{m+n} \right)$$
• Midpoint and centroid · Centroid of triangle ABC
  $$G = \left( \frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3},\ \frac{z_1+z_2+z_3}{3} \right)$$

Reference tables (1)

Watch out for (3)

• Don't forget the square root — or any one of the three squared terms
  → Distance between two points
• The ratio m : n weights the FAR endpoint by m — don't swap the weights
  → Section formula — dividing a segment in a ratio
• Collinear vs coplanar vs concyclic
  → Collinearity and shape tests

Formulas (5)

• Direction ratios, direction cosines, and the unit identity · Direction cosines square-sum to 1
  $$l^2 + m^2 + n^2 = 1, \quad l = \cos\alpha,\ m = \cos\beta,\ n = \cos\gamma$$
• Angle between two lines · Angle between two lines (direction ratios)
  $$\cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$$
• Perpendicular and parallel conditions · Perpendicularity condition
  $$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$$
• Projection of a segment on an axis or line · Projection of AB on a line of direction cosines ⟨l, m, n⟩
  $$\text{proj} = (x_2-x_1)\,l + (y_2-y_1)\,m + (z_2-z_1)\,n$$
• Direction-angle identities · Core identities
  $$\sum \cos^2\!\theta = 1, \quad \sum \sin^2\!\theta = 2, \quad \sum \cos 2\theta = -1$$

Reference tables (1)

Watch out for (5)

• Direction RATIOS are not unique; direction COSINES (almost) are
  → Direction ratios, direction cosines, and the unit identity
• You must NORMALISE direction ratios into cosines — and the identity is = 1, not = 0
  → Direction ratios, direction cosines, and the unit identity
• Mind the coefficient and the sign before reading denominators
  → Reading direction ratios off a line
• For two LINES use their DIRECTIONS, not normals; divide by BOTH magnitudes
  → Angle between two lines
• A line cannot make equal acute angles with all three axes unless it's the diagonal
  → Direction-angle identities

Formulas (2)

• Equation of a line — symmetric and two-point forms · Symmetric form of a line
  $$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} = t$$
• Line parallel to, or lying in, a plane · Parallel/contained condition (direction ⟂ normal)
  $$A a + B b + C c = 0$$

Watch out for (2)

• Parallel vs lying-in needs the second check
  → Line parallel to, or lying in, a plane
• direction · normal = 0 means the line is PARALLEL to the plane, not perpendicular
  → Line parallel to, or lying in, a plane

Formulas (6)

• Equation of a plane and its normal · Point-normal form
  $$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$$
• Intercept form and special planes · Intercept form
  $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$
• Plane through three points · Determinant form through three points
  $$\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$$
• Distance from a point and the foot of the perpendicular · Distance from a point to a plane
  $$\text{distance} = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$
• Angle between two planes · Angle between planes (via normals)
  $$\cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}$$
• Plane through the line of intersection of two planes · Pencil of planes
  $$P_1 + \lambda P_2 = 0$$

Watch out for (2)

• Scale parallel planes to a common normal BEFORE subtracting constants
  → Distance from a point and the foot of the perpendicular
• Always divide by $\sqrt{a^2+b^2+c^2}$ — the numerator alone is NOT the distance
  → Distance from a point and the foot of the perpendicular

Formulas (4)

• General equation, centre, and radius · Centre and radius of a sphere
  $$\text{centre} = (-u, -v, -w), \quad r = \sqrt{u^2 + v^2 + w^2 - d}$$
• Diameter form of a sphere · Diameter form
  $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) + (z-z_1)(z-z_2) = 0$$
• Sphere and a plane — tangency and sections · Tangency condition
  $$p = r \quad\text{(perpendicular distance from centre = radius)}$$
• Sphere and the coordinate axes · Distance from a point to the z-axis
  $$\text{dist to } z\text{-axis} = \sqrt{x_c^2 + y_c^2}$$

Watch out for (3)

• The centre is $(-u, -v, -w)$ — NEGATE the half-coefficients
  → General equation, centre, and radius
• Radius is $\sqrt{u^2+v^2+w^2-d}$ — mind the $-d$ sign and don't drop the $\sqrt{}$
  → General equation, centre, and radius
• Touching a PLANE vs touching an AXIS
  → Sphere and a plane — tangency and sections