Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).
In the Cartesian or symmetrical form of equation the angle between two lines can be found by dot product equation and in this equation we will use the direction ratios which are in the denominator of the equation. 
in this equation a,b,c are the direction ratios of this equation.
Direction ratios of the first line are a1 = 2, b1 = 2, c1 = 1 which corresponds to 2,2,1.
Direction ratios of the line joining (3,1,4) and (7,2,12) .
= (7–3,2–1,12–4) = (4,1,8) .
a2 = 4, b2 = 1, c2 = 8
now to find the angle between two lines we use cross product equation,
cos θ = 
θ = 
11