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 l2 + m2 + n2 = 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 (x1, y1, z2) and B (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1) (or) (x1 – x2, y1 – y2, z1 – z2)
→ 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 (a1, b1, c1) and (a2, b2, c2) 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 a1a2 + b1b2 + c1c2 = 0.
→ In terms of direction cosine’s cos θ = l1l2 + m1m2 + n1n2, and for perpendicular lines l1l2 + m1m2 + n1n2 = 0.