Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.

Given that the direction ratios of the lines are proportional to 2:3:6 and 1:2:2.

Let us denote the lines in the form of vectors as A and B.

Let’s write the vectors:

⇒ A = 2i + 3j + 6k

⇒ B = 1i + 2j + 2k

We know that the angle between the vectors a₁i + b₁j + c₁k and a₂i + b₂j + c₂k is given by:

⇒

Let’s assume the angle between the vectors A and B be ,

Using the given formula we find the value of .

⇒

⇒

⇒

⇒

⇒

The acute angle between the two vectors is given by .