## Three - dimensional Geometry

# Direction Cosines and Direction Ratios of a Line

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

i.e., cos^{2}α + cos^{2}β + cos^{2}γ = 1

i.e., sin^{2}α + sin^{2}β + sin^{2}γ = 2 - If A(a
_{1}, b_{1}, c_{1}) B(a_{2}, b_{2}, c_{2}), then direction ratios of a line joining points A and B is = (a_{2}− a_{1}, b_{2}− b_{1}, c_{2}− c_{1}) - If a, b, c are the direction ratios of the line then its direction cosines are \left(\frac{\pm a}{\sqrt{a^{2}+b^{2}+c^{2}}},\frac{\pm b}{\sqrt{a^{2}+b^{2}+c^{2}}},\frac{\pm c}{\sqrt{a^{2}+b^{2}+c^{2}}}\right)
- If l, m, n are the direction cosines of the line then its direction ratios are = (kl, km, kn) ∀ k ∈ R

### View the Topic in this video From 22:50 To 31:18

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

2. Direction cosines of a line joining two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are \frac{x_{2}-x_{1}}{PQ},\frac{y_{2}-y_{1}}{PQ},\frac{z_{2}-z_{1}}{PQ}

where PQ=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}

3. If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then 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}}}