Learning these TS Inter 1st Year Maths 1B Formulas Chapter 6 Direction Cosines and Direction Ratios will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1B Direction Cosines and Direction Ratios Formulas

→ If a line makes angles a, [3. y with the coordinate axes then cos α, cos β, cos γ are called ‘the direction cosines of the lines denoted by l, m, n.

The relation between l, in. n is l^{2} + m^{2} + n^{2} = 1

→ An ordered triple of numbers proportional to the direction cosines of a line are called as direction ratios of the line.

→ If a, b, c are the dirrc!ion ratios of a ray then the direction cosine are given by \(\left(\frac{a}{\sqrt{a^2+b^2}+c^2} \cdot \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2}+c^2}\right)\)

→ Direction ratios of the line joining A (x_{1}, y_{1}, z_{2}) and B (x_{2}, y_{2}, z_{2}) are (x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}) (or) (x_{1} – x_{2}, y_{1} – y_{2}, z_{1} – z_{2})

→ Direction cosines of the above line = \(\left(\frac{x_2-x_1}{A B}, \frac{y_2-y_1}{A B}, \frac{z_2-z_1}{A B}\right)\)

→ If θ is the angle between two lines with direction ratio’s (a_{1}, b_{1}, c_{1}) and (a_{2}, b_{2}, c_{2}) then

cos θ = \(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{\left(a_1^2+b_1^2+c_1^2\right)\left(a_2^2+b_2^2+c_2^2\right)}}\)

→ If the above lines are perpendicular then a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0.

→ In terms of direction cosine’s cos θ = l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}, and for perpendicular lines l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2} = 0.