The direction cosines are obtained by dividing the direction ratios by the square root of the sum of their squares. Here the magnitude = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3. So the direction cosines are l = 1/3, m = 2/3 and n = 2/3. We can check l^2 + m^2 + n^2 = 1/9 + 4/9 + 4/9 = 9/9 = 1.

l = a/sqrt(a^2+b^2+c^2), etc.

Vector form: the equation of a line passing through a point with position vector a and parallel to a vector b is r = a + lambda b, where lambda is a scalar parameter and r is the position vector of any point on the line. Cartesian form: if the line passes through the point (x1, y1, z1) and has direction ratios a, b, c, then its equation is (x - x1)/a = (y - y1)/b = (z - z1)/c.

r = a + lambda b

If two lines have direction ratios (a1, b1, c1) and (a2, b2, c2), the angle theta between them is given by cos theta = |a1 a2 + b1 b2 + c1 c2| / (sqrt(a1^2 + b1^2 + c1^2) x sqrt(a2^2 + b2^2 + c2^2)). The lines are perpendicular if a1 a2 + b1 b2 + c1 c2 = 0, and they are parallel if a1/a2 = b1/b2 = c1/c2 (their direction ratios are proportional).

cos theta = |a1a2 + b1b2 + c1c2|/(|d1||d2|)