Direction Ratios, Direction Cosine & Direction Angles

The angles made by a vector with the positive directions of the X-axis, Y-axis and Z-axis are called direction angles of the vector, denoted by α, β, and γ.

If α, β and γ are the direction angles of a vector, then the cosines of these angles, i.e.

l = cos⁡α, m = cos⁡β, n = cos⁡γ 

are called the direction cosines of the vector.

If point is (x,y,z)and distance r: \[\cos\alpha=\frac{x}{r},\quad\cos\beta=\frac{y}{r},\quad\cos\gamma=\frac{z}{r}\]

If l, m, n are direction cosines of a line and if a, b, c are real numbers such that \[\frac{\mathrm{a}}{l}=\frac{\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{c}}{\mathrm{n}}=\lambda,\] then a, b, c are called direction ratios of that line.

1. Conversion: From D.R → D.C: \[l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]

2. Angle between two lines

If direction cosines: \[\cos\theta=l_1l_2+m_1m_2+n_1n_2\]

If direction ratios: \[\cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\]

3. If A(x₁, y₁, z₁), B(x₂, y₂, z₂):

\[\mathrm{D.Rs}=(x_2-x_1,y_2-y_1,z_2-z_1)\]

4. \[l^2+m^2+n^2=1\]