**Introduction**

To find the direction cosines of a line perpendicular to two given lines, we can make use of the fact that the dot product of two perpendicular vectors is zero. By finding the dot product of the direction cosines of the line in question with the given lines, we can solve for the unknown direction cosines.

**Given Information**

We are given two lines with direction cosines proportional to (1, -2, -2) and (0, 2, 1). Let's denote the direction cosines of the unknown line as (l, m, n).

**Finding the Dot Product**

To find the direction cosines of the unknown line, we need to find the dot product of its direction cosines with the direction cosines of the given lines. The dot product is calculated as follows:

(l, m, n) · (1, -2, -2) = l + (-2m) + (-2n) = 0 (1)

(l, m, n) · (0, 2, 1) = 0 + (2m) + n = 0 (2)

**Solving the Equations**

We now have two equations (1) and (2) with two unknowns (l, m, n). Let's solve these equations simultaneously.

From equation (2), we can express n in terms of m:

n = -2m (3)

Substituting equation (3) into equation (1), we get:

l + (-2m) + (-2(-2m)) = 0

l + (-2m) + 4m = 0

l + 2m = 0

l = -2m (4)

**Finding the Direction Cosines**

Now that we have an expression for l in terms of m, we can express the direction cosines of the unknown line as:

(l, m, n) = (-2m, m, -2m)

To find the direction cosines, we need to normalize this vector by dividing each component by its magnitude. The magnitude of the vector is:

√((-2m)^2 + m^2 + (-2m)^2) = √(4m^2 + m^2 + 4m^2) = √(9m^2) = 3|m|

Dividing each component by 3|m|, we get:

(-2m/3|m|, m/3|m|, -2m/3|m|)

These are the direction cosines of the line perpendicular to the given lines with direction cosines proportional to (1, -2, -2) and (0, 2, 1).

**Conclusion**

To find the direction cosines of a line perpendicular to two given lines, we can use the fact that the dot product of two perpendicular vectors is zero. By finding the dot product of the direction cosines of the unknown line with the direction cosines of the given lines, we can solve for the unknown direction cosines. In this case, the direction cosines of the line perpendicular to the given lines with direction cosines proportional to (1, -2, -2) and (0, 2, 1) are given by (-2m/3|m|, m/3|m|, -2m/3|m|).

Direction cosines are ( 2/3 , -1/3, 2/3)