Any multiples of direction cosines. If direction ratios are a, b, c then:
l = a/sqrt(a^2+b^2+c^2), m = b/sqrt(a^2+b^2+c^2), n = c/sqrt(a^2+b^2+c^2)
6. Equation of a Line through point (x1, y1, z1) with direction ratios a, b, c:
(x - x1)/a = (y - y1)/b = (z - z1)/c
7. Angle between two lines with direction cosines (l1,m1,n1) and (l2,m2,n2):
cos(theta) = |l1*l2 + m1*m2 + n1*n2|
8.
Distance from Origin to point (x, y, z):
r = sqrt(x^2 + y^2 + z^2)
FORMULA BLOCK
- l^2 + m^2 + n^2 = 1 (Always true for direction cosines)
- If lines are perpendicular: l1*l2 + m1*m2 + n1*n2 = 0
- If lines are parallel: l1/l2 = m1/m2 = n1/n2
EXAM PATTERNS
NDA regularly asks: (1) Finding distance between two 3D points, (2) Direction cosines of a line, (3) Verifying if two lines are perpendicular or parallel, (4) Finding coordinates of a dividing point. Most questions are direct formula-based. If you know the formula, you can solve in under 60 seconds.
SHORTCUT TRICK
Trick 1 — Direction Cosine Check: If you get direction cosines from a problem, always verify l^2 + m^2 + n^2 = 1. If it does not equal 1, your answer is wrong.
Use this as instant self-check