# Direction Cosines and Direction Ratios of a Line

• If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1.
i.e., cos2 α + cos2β + cos2γ = 1
i.e., sin2 α + sin2β + sin2γ = 2
• If A(a1, b1, c1) B(a2, b2, c2), then direction ratios of a line joining points A and B is = (a2 − a1, b2 − b1, c2 − c1)
• 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 l2 + m2 + n2 = 1.

2. Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) 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}}}