Finding Direction Ratios of Perpendicular Lines

We are asked to find the direction ratios of a line that is perpendicular to two given lines. The direction ratios of the two given lines are $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$.

If a line is perpendicular to two lines with direction ratios $\langle a_1, b_1, c_1 \rangle$ and $\langle a_2, b_2, c_2 \rangle$, then the direction ratios of the perpendicular line, $\langle a, b, c \rangle$, are proportional to the cross product of the direction vectors corresponding to the given direction ratios.

The cross product of two vectors $\mathbf{v}_1 = a_1\mathbf{i} + b_1\mathbf{j} + c_1\mathbf{k}$ and $\mathbf{v}_2 = a_2\mathbf{i} + b_2\mathbf{j} + c_2\mathbf{k}$ is given by:

$\mathbf{v}_1 \times \mathbf{v}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = (b_1c_2 - b_2c_1)\mathbf{i} - (a_1c_2 - a_2c_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$

The direction ratios of the resulting vector are $\langle (b_1c_2 - b_2c_1), -(a_1c_2 - a_2c_1), (a_1b_2 - a_2b_1) \rangle$, which is equivalent to $\langle (b_1c_2 - b_2c_1), (a_2c_1 - a_1c_2), (a_1b_2 - a_2b_1) \rangle$.

Given direction ratios:

• Line 1: $\langle a_1, b_1, c_1 \rangle = \langle 1, -2, -2 \rangle$
• Line 2: $\langle a_2, b_2, c_2 \rangle = \langle 0, 2, 1 \rangle$

Let the direction ratios of the perpendicular line be $\langle a, b, c \rangle$. We can calculate these using the cross product formula:

• $a = b_1c_2 - b_2c_1 = (-2)(1) - (2)(-2) = -2 - (-4) = -2 + 4 = 2$
• $b = c_1a_2 - c_2a_1 = (-2)(0) - (1)(1) = 0 - 1 = -1$
• $c = a_1b_2 - a_2b_1 = (1)(2) - (0)(-2) = 2 - 0 = 2$

So, the direction ratios of the line perpendicular to the given lines are proportional to $\langle 2, -1, 2 \rangle$.

Now, let's compare this with the given options:

1. $\langle 2, -1, 2 \rangle$
2. $\langle -2, 1, 2 \rangle$
3. $\langle 2, 1, -2 \rangle$
4. $\langle -2, -1, -2 \rangle$

The calculated direction ratios $\langle 2, -1, 2 \rangle$ match the first option.

Conclusion

The direction ratios of the line perpendicular to the lines with direction ratios $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$ are $\langle 2, -1, 2 \rangle$.