Direction cosines from ratios
\[l=\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\frac{b}{\sqrt{a^2+b^2+c^2}},\ n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]
Three Dimensional Geometry
Direction Cosines and Direction Ratios
Understand
If a directed line makes angles \(\alpha,\beta,\gamma\) with the positive \(x\), \(y\), \(z\) axes, then \(l=\cos\alpha\), \(m=\cos\beta\), \(n=\cos\gamma\) are its direction cosines, and
\[l^2+m^2+n^2=1.\]
Any three numbers \(a,b,c\) proportional to \(l,m,n\) are direction ratios. From direction ratios: \(l=\frac{a}{\sqrt{a^2+b^2+c^2}}\) (and similarly \(m,n\)), with a choice of sign giving the two opposite directions.
The direction ratios of the line through \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) are \(x_2-x_1,\ y_2-y_1,\ z_2-z_1\).
The coordinate axes have direction cosines \(1,0,0\) (\(x\)-axis), \(0,1,0\) and \(0,0,1\).
Key Concepts
- l² + m² + n² = 1.
- Direction ratios: any multiple of (l, m, n).
- DRs of PQ: differences of coordinates.
Formula Bank
Common Mistakes
DRs are not DCs
\(2,-1,2\) are direction ratios; the direction cosines are \(\frac23,-\frac13,\frac23\).
Practice & Topic Test
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Direction ratios come from coordinate differences; divide by their length to get direction cosines.
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