If a line has direction ratios proportional to 2, -1, -2, then what are its direction consines?

Given, the direction ratios of the line are proportional to (2, -1,-2)

Therefore, consider the direction ratios of the give line can be

a = 2 × k, b = (-1) × k, c = (-2) × k

[where k is some proportionality constant]

Now the direction ratios of the line are

a = 2k, b = -k, c = -2k

As we know the direction cosine ae given by

cos α = l , cos β = m , cos γ = n

Where α, β and γ are the angles formed by the line with the three axes.

By using the above formula:

l = cos α =

Therefore cos α

m =cos β =

cos β

n = cos γ =

cos γ

Therefore, the direction cosines are