Three Dimensional Geometry — Formula Sheet

l^2 + m^2 + n^2 = 1

l = ± a / √(a^2 + b^2 + c^2), m = ± b / √(a^2 + b^2 + c^2), n = ± c / √(a^2 + b^2 + c^2)

a = λl, b = λm, c = λn, where λ ≠ 0

Direction cosines of the line through P(x1, y1, z1) and Q(x2, y2, z2): (x2 - x1)/PQ, (y2 - y1)/PQ, (z2 - z1)/PQ

Direction ratios of PQ: x2 - x1, y2 - y1, z2 - z1 or x1 - x2, y1 - y2, z1 - z2

r = a + λb

(x - x1)/a = (y - y1)/b = (z - z1)/c

x = x1 + λa, y = y1 + λb, z = z1 + λc

cos θ = |(a1a2 + b1b2 + c1c2) / (√(a1^2 + b1^2 + c1^2) √(a2^2 + b2^2 + c2^2))|

sin θ = √((a1b2 - a2b1)^2 + (b1c2 - b2c1)^2 + (c1a2 - c2a1)^2) / √(a1^2 + b1^2 - c1^2) √(a2^2 + b2^2 + c2^2)

cos θ = |l1l2 + m1m2 + n1n2|

sin θ = √((l1m2 - l2m1)^2 - (m1n2 - m2n1)^2 + (n1l2 - n2l1)^2)

cos θ = |b1 · b2| / (|b1| |b2|)

d = |((b1 × b2) · (a2 - a1)) / |b1 × b2||

d = | determinant | / √((b1c2 - b2c1)^2 + (c1a2 - c2a1)^2 + (a1b2 - a2b1)^2)

d = |b × (a2 - a1)| / |b|

If the direction cosines of a line are l, m, n, then l^2 + m^2 + n^2 = 1.

Cartesian form

(x-x1)/a = (y-y1)/b = (z-z1)/c.