3d Geometry

To find the direction cosines of a line perpendicular to the given lines, we need to find a vector that is perpendicular to both vectors represented by the direction ratios (1, -2, -2) and (0, 2, 1). We achieve this by computing the cross product of these two vectors:




The cross product is given by:





Thus, the direction ratios of the perpendicular line are (2, -1, 2).

Now, we convert these direction ratios to direction cosines by dividing each value by the magnitude of the vector.

Magnitude of vector:

So, the direction cosines are:


Thus, the correct option is .

The angle between two lines whose direction ratios are given by and can be found using the formula:

First, compute the dot product :

Next, compute the magnitudes of and :

Thus, we have:

Therefore,

Thus, the correct answer is: .