Find the direction of a line segment whose direction ratios are:
2,-6, 3
2, -1, -2
`-9,6,-2`

Explanation

To find the direction cosines of a line segment given its direction ratios, we follow these steps:

### Step 1: Identify the Direction Ratios
The direction ratios given are:
1. \( \langle 2, -6, 3 \rangle \)
2. \( \langle 2, -1, -2 \rangle \)
3. \( \langle -9, 6, -2 \rangle \)

### Step 2: Calculate the Magnitude of Each Direction Ratio
The magnitude of a vector \( \langle a, b, c \rangle \) is given by:
\[ \text{Magnitude} = \sqrt{a^2 + b^2 + c^2} \]

#### For the first direction ratio \( \langle 2, -6, 3 \rangle \):
\[ \text{Magnitude} = \sqrt{2^2 + (-6)^2 + 3^2} = \sqrt{4 + 36 + 9} = \sqrt{49} = 7 \]

#### For the second direction ratio \( \langle 2, -1, -2 \rangle \):
\[ \text{Magnitude} = \sqrt{2^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

#### For the third direction ratio \( \langle -9, 6, -2 \rangle \):
\[ \text{Magnitude} = \sqrt{(-9)^2 + 6^2 + (-2)^2} = \sqrt{81 + 36 + 4} = \sqrt{121} = 11 \]

### Step 3: Calculate the Direction Cosines
The direction cosines are calculated by dividing each component of the direction ratio by its magnitude.

#### For the first direction ratio \( \langle 2, -6, 3 \rangle \):
\[ \text{Direction Cosines} = \left( \frac{2}{7}, \frac{-6}{7}, \frac{3}{7} \right) \]

#### For the second direction ratio \( \langle 2, -1, -2 \rangle \):
\[ \text{Direction Cosines} = \left( \frac{2}{3}, \frac{-1}{3}, \frac{-2}{3} \right) \]

#### For the third direction ratio \( \langle -9, 6, -2 \rangle \):
\[ \text{Direction Cosines} = \left( \frac{-9}{11}, \frac{6}{11}, \frac{-2}{11} \right) \]

### Final Answer
The direction cosines for the three sets of direction ratios are:
1. \( \left( \frac{2}{7}, \frac{-6}{7}, \frac{3}{7} \right) \)
2. \( \left( \frac{2}{3}, \frac{-1}{3}, \frac{-2}{3} \right) \)
3. \( \left( \frac{-9}{11}, \frac{6}{11}, \frac{-2}{11} \right) \)