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3-D Line: Direction Ratios → Cosines

Three-Dimensional Geometry · Class XII

Set direction ratios (a, b, c) of a 3-D line. Convert to direction cosines (l, m, n).

2
type a value
-55
3
type a value
-55
6
type a value
-55
Live values
  • Magnitude7
  • l0.2857
  • m0.4286
  • n0.8571
  • Check l² + m² + n²1
xy
  • Magnitude as c slides

Formulas in this lab

  • Magnitude
    a2+b2+c2\sqrt{a^2+b^2+c^2}
  • l
    a/a2+b2+c2a/\sqrt{a^2+b^2+c^2}
  • m
    b/a2+b2+c2b/\sqrt{a^2+b^2+c^2}
  • n
    c/a2+b2+c2c/\sqrt{a^2+b^2+c^2}
  • Check l² + m² + n²
    =1= 1
Tip: Magnitude is a hyperbola in c; minimum sqrt(a^2 + b^2) at c = 0, then grows linearly for large |c|.

Frequently asked questions

▶What are direction ratios and direction cosines?

Direction ratios $(a, b, c)$ describe a line's direction (not magnitude). Direction cosines $(l, m, n) = (a, b, c)/\sqrt{a^2+b^2+c^2}$ are normalized to unit length and satisfy $l^2 + m^2 + n^2 = 1$.

▶How do I use the direction ratios lab?

Slide $(a, b, c)$. The lab returns the magnitude $\sqrt{a^2+b^2+c^2}$, the direction cosines $(l, m, n)$ and verifies $l^2 + m^2 + n^2 = 1$. Try $(2, 3, 6)$: magnitude $7$, $(l, m, n) = (2/7, 3/7, 6/7)$.

▶Why must $l^2 + m^2 + n^2 = 1$ always hold?

Because $(l, m, n)$ is a unit vector by construction. JEE asks: are $(1/2, 1/2, 1/2)$ valid direction cosines? Check: $1/4 + 1/4 + 1/4 = 3/4 \ne 1$, so no. The lab does this check automatically.

▶Where do direction cosines show up beyond maths?

Robotics encodes the orientation of a robot arm using direction cosines (or quaternions, which are equivalent). Crystallography uses them for crystal-plane orientations. Even the autopilot in a Boeing 787 uses direction cosines to track heading.