The direction cosines of the line joining the points `(4,3,-5)` and `(-2,1-8)` are

Options

A. `lt2,4,-13gt`
B. `lt6,2,3 gt`
C. `lt6//7,2//7,3//7gt`
D. None of these

Answer

To find the direction cosines of the line joining the points \( A(4, 3, -5) \) and \( B(-2, 1, -8) \), we will follow these steps:

### Step 1: Find the direction ratios of the line
The direction ratios of the line joining two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) can be calculated using the formula:
\[ (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ x_2 - x_1 = -2 - 4 = -6 \]
\[ y_2 - y_1 = 1 - 3 = -2 \]
\[ z_2 - z_1 = -8 - (-5) = -3 \]
Thus, the direction ratios are \( (-6, -2, -3) \).

### Step 2: Calculate the magnitude of the direction ratios
The magnitude of the direction ratios can be calculated using the formula:
\[ \text{Magnitude} = \sqrt{(-6)^2 + (-2)^2 + (-3)^2} \]
Calculating each term:
\[ (-6)^2 = 36, \quad (-2)^2 = 4, \quad (-3)^2 = 9 \]
Adding these values:
\[ 36 + 4 + 9 = 49 \]
Taking the square root:
\[ \text{Magnitude} = \sqrt{49} = 7 \]

### Step 3: Find the direction cosines
The direction cosines \( l, m, n \) can be found by dividing each direction ratio by the magnitude:
\[ l = \frac{-6}{7}, \quad m = \frac{-2}{7}, \quad n = \frac{-3}{7} \]
Thus, the direction cosines are:
\[ \left( -\frac{6}{7}, -\frac{2}{7}, -\frac{3}{7} \right) \]

### Step 4: Consider the positive values
Direction cosines can also be represented in positive form. Therefore, the direction cosines can also be:
\[ \left( \frac{6}{7}, \frac{2}{7}, \frac{3}{7} \right) \]

### Final Answer
The direction cosines of the line joining the points \( (4, 3, -5) \) and \( (-2, 1, -8) \) are:
\[ \left( \frac{6}{7}, \frac{2}{7}, \frac{3}{7} \right) \quad \text{and} \quad \left( -\frac{6}{7}, -\frac{2}{7}, -\frac{3}{7} \right) \]