Three-Dimensional Geometry • Topic 1 of 3

Direction Cosines & Direction Ratios

A line in space has a direction. The direction cosines of a line are the cosines of the angles $\alpha,\beta,\gamma$ that it makes with the positive $x$, $y$, $z$ axes, written $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$.

The fundamental identity

Direction cosines always satisfy

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

Direction ratios

Any three numbers $a,b,c$ proportional to $l,m,n$ are called direction ratios. They are not unique (any non-zero multiple works). To convert ratios to cosines, divide by the magnitude:

$$l=\frac{a}{\sqrt{a^2+b^2+c^2}},\quad m=\frac{b}{\sqrt{a^2+b^2+c^2}},\quad n=\frac{c}{\sqrt{a^2+b^2+c^2}}.$$

Line through two points

The direction ratios of the line through $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ are $x_2-x_1,\ y_2-y_1,\ z_2-z_1$.

Solved Examples
1
Worked Example
Find the direction cosines of a line with direction ratios $1,2,2$.
Solution

Magnitude $=\sqrt{1+4+4}=3$. So $l=\tfrac13,\ m=\tfrac23,\ n=\tfrac23$. Check $l^2+m^2+n^2=\tfrac{1+4+4}{9}=1.$

2
Worked Example
Find direction ratios of the line joining $A(1,2,3)$ and $B(4,5,6)$.
Solution

$(4-1,5-2,6-3)=(3,3,3)$, or simplified $(1,1,1)$.

3
Worked Example
If a line makes equal angles with the axes, find its direction cosines.
Solution

$l=m=n$. From $3l^2=1$, $l=\pm\dfrac{1}{\sqrt3}$. So the direction cosines are $\left(\tfrac{1}{\sqrt3},\tfrac{1}{\sqrt3},\tfrac{1}{\sqrt3}\right)$.

4
Worked Example
Can $\tfrac12,\tfrac12,\tfrac12$ be direction cosines of a line?
Solution

Check $l^2+m^2+n^2=\tfrac14+\tfrac14+\tfrac14=\tfrac34\ne1$. So no — they fail the fundamental identity.

Key Points

  • Direction cosines $l,m,n=\cos\alpha,\cos\beta,\cos\gamma$ with $l^2+m^2+n^2=1$.
  • Direction ratios are any numbers proportional to $l,m,n$ (not unique).
  • Convert ratios $a,b,c$ to cosines by dividing by $\sqrt{a^2+b^2+c^2}$.
  • Line through two points: DRs are the coordinate differences.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Direction cosines satisfy:
Explanation: The fundamental identity.
Q2.Direction cosines of a line with DRs $1,2,2$ are:
Explanation: Divide by magnitude $3$.
Q3.DRs of the line joining $(1,2,3),(4,5,6)$ are:
Explanation: Coordinate differences.
Q4.A line equally inclined to all axes has each direction cosine:
Explanation: $3l^2=1\Rightarrow l=1/\sqrt3$.
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