Calculate the angle between lines AB and CD if A (√3; 1; 0), B (8; -2; 4), C (0; 2; 0), D (√3,1; 2√2).

To calculate the angle between straight lines, we will use the formula for finding the angle between vectors: cos (A) = AB * CD / | AB | * | CD |. For this we represent the segments AB and CD as vectors specified by the coordinates of their ends. Let’s calculate the coordinates of these vectors:

AB = ((0 – 1); (1 – 1); (1 – 2)) = (-1; 0; -1); CD = ((2 – 2); (-3 – 2); (1 – 2)) = (0; -5; -1).

Let’s find their modules:

| AB | = √ ((- 1) ^ 2 + 0 ^ 2 + (-1) ^ 2) = √ (1 + 1) = √2;

| CD | = √ (0 ^ 2 + (-5) ^ 2 + (-1) ^ 2) = √ (25 + 1) = √26.

Now:

cos (A) = AB * CD / | AB | * | CD | = ((-1) * 0 + 0 * (-5) + (-1) * (-1)) / (√2 * √26) = 1 / √52 = 1 / 2√13.

A = arccos (1 / 2√13).

Answer: the angle between the straight lines is A = arccos (1 / 2√13).