Find the direction cosines of the line joining the following points :
(i) `A(2,-1,3), B(3,1.1)`
( ii) `A(2,-1,2), B(-4,2,0)`
(iii) `A(4,3,-5), B(-2,1,-8)`

Answer

To find the direction cosines of the line joining two points in three-dimensional space, we follow these steps:

### Step-by-Step Solution

**(i) For points A(2, -1, 3) and B(3, 1, 1):**

1. **Find the direction ratios:**
- Direction ratios are given by the differences in coordinates:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
- Here, \(x_1 = 2\), \(y_1 = -1\), \(z_1 = 3\) and \(x_2 = 3\), \(y_2 = 1\), \(z_2 = 1\).
- Thus,
\[ \text{Direction Ratios} = (3 - 2, 1 - (-1), 1 - 3) = (1, 2, -2) \]

2. **Calculate the magnitude of the direction ratios:**
- The magnitude \(d\) is given by:
\[ d = \sqrt{(1^2 + 2^2 + (-2)^2)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

3. **Find the direction cosines:**
- The direction cosines are calculated as:
\[ \left( \frac{x_2 - x_1}{d}, \frac{y_2 - y_1}{d}, \frac{z_2 - z_1}{d} \right) \]
- Therefore,
\[ \text{Direction Cosines} = \left( \frac{1}{3}, \frac{2}{3}, \frac{-2}{3} \right) \]

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**(ii) For points A(2, -1, 2) and B(-4, 2, 0):**

1. **Find the direction ratios:**
- Here, \(x_1 = 2\), \(y_1 = -1\), \(z_1 = 2\) and \(x_2 = -4\), \(y_2 = 2\), \(z_2 = 0\).
- Thus,
\[ \text{Direction Ratios} = (-4 - 2, 2 - (-1), 0 - 2) = (-6, 3, -2) \]

2. **Calculate the magnitude of the direction ratios:**
- The magnitude \(d\) is given by:
\[ d = \sqrt{((-6)^2 + 3^2 + (-2)^2)} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7 \]

3. **Find the direction cosines:**
- Therefore,
\[ \text{Direction Cosines} = \left( \frac{-6}{7}, \frac{3}{7}, \frac{-2}{7} \right) \]

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**(iii) For points A(4, 3, -5) and B(-2, 1, -8):**

1. **Find the direction ratios:**
- Here, \(x_1 = 4\), \(y_1 = 3\), \(z_1 = -5\) and \(x_2 = -2\), \(y_2 = 1\), \(z_2 = -8\).
- Thus,
\[ \text{Direction Ratios} = (-2 - 4, 1 - 3, -8 - (-5)) = (-6, -2, -3) \]

2. **Calculate the magnitude of the direction ratios:**
- The magnitude \(d\) is given by:
\[ d = \sqrt{((-6)^2 + (-2)^2 + (-3)^2)} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

3. **Find the direction cosines:**
- Therefore,
\[ \text{Direction Cosines} = \left( \frac{-6}{7}, \frac{-2}{7}, \frac{-3}{7} \right) \]

### Summary of Direction Cosines:

1. For points A(2, -1, 3) and B(3, 1, 1): \(\left( \frac{1}{3}, \frac{2}{3}, \frac{-2}{3} \right)\)
2. For points A(2, -1, 2) and B(-4, 2, 0): \(\left( \frac{-6}{7}, \frac{3}{7}, \frac{-2}{7} \right)\)
3. For points A(4, 3, -5) and B(-2, 1, -8): \(\left( \frac{-6}{7}, \frac{-2}{7}, \frac{-3}{7} \right)\)

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