Find the direction cosines of the line passing through (0, 1, -2) and (-2, 3, 6).

Answer

To find the direction cosines of the line passing through the points \( P(0, 1, -2) \) and \( Q(-2, 3, 6) \), we will follow these steps:

### Step 1: Identify the coordinates of the points
Let:
- \( P(x_1, y_1, z_1) = (0, 1, -2) \)
- \( Q(x_2, y_2, z_2) = (-2, 3, 6) \)

### Step 2: Calculate the differences in coordinates
We need to find the differences in the coordinates:
- \( x_2 - x_1 = -2 - 0 = -2 \)
- \( y_2 - y_1 = 3 - 1 = 2 \)
- \( z_2 - z_1 = 6 - (-2) = 6 + 2 = 8 \)

### Step 3: Calculate the distance \( PQ \)
The distance \( PQ \) between the two points is given by the formula:
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the values we calculated:
\[ PQ = \sqrt{(-2)^2 + (2)^2 + (8)^2} \]
\[ = \sqrt{4 + 4 + 64} = \sqrt{72} \]
\[ = 6\sqrt{2} \]

### Step 4: Calculate the direction cosines
The direction cosines \( l, m, n \) are given by:
\[ l = \frac{x_2 - x_1}{PQ}, \quad m = \frac{y_2 - y_1}{PQ}, \quad n = \frac{z_2 - z_1}{PQ} \]
Substituting the values:
\[ l = \frac{-2}{6\sqrt{2}}, \quad m = \frac{2}{6\sqrt{2}}, \quad n = \frac{8}{6\sqrt{2}} \]

### Step 5: Simplify the direction cosines
Now we simplify each of these:
\[ l = \frac{-2}{6\sqrt{2}} = \frac{-1}{3\sqrt{2}} = \frac{-\sqrt{2}}{6} \]
\[ m = \frac{2}{6\sqrt{2}} = \frac{1}{3\sqrt{2}} = \frac{\sqrt{2}}{6} \]
\[ n = \frac{8}{6\sqrt{2}} = \frac{4}{3\sqrt{2}} = \frac{2\sqrt{2}}{3} \]

### Final Result
Thus, the direction cosines of the line are:
\[ \left( \frac{-\sqrt{2}}{6}, \frac{\sqrt{2}}{6}, \frac{2\sqrt{2}}{3} \right) \]