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

Options

A. `-3,1,4`
B. `-1,5,-2`
C. `(-3)/(sqrt(26)),(1)/(sqrt(26)),(4)/(sqrt(26))`
D. `(-1)/(sqrt(30)),(5)/(sqrt(30)),(-2)/(sqrt(30))`

Answer

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

### Step 1: Determine the coordinates of the points
The coordinates of point \( A \) are \( (1, 2, -3) \) and the coordinates of point \( B \) are \( (-2, 3, 1) \).

### Step 2: Calculate the direction ratios
The direction ratios (DR) 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:
\[ \text{DR} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ \text{DR} = (-2 - 1, 3 - 2, 1 - (-3)) = (-3, 1, 4) \]

### Step 3: Calculate the magnitude of the direction ratios
The magnitude \( d \) of the direction ratios is given by:
\[ d = \sqrt{(-3)^2 + 1^2 + 4^2} \]
Calculating this:
\[ d = \sqrt{9 + 1 + 16} = \sqrt{26} \]

### Step 4: Calculate the direction cosines
The direction cosines \( l, m, n \) can be calculated using the formulas:
\[ l = \frac{a}{d}, \quad m = \frac{b}{d}, \quad n = \frac{c}{d} \]
where \( (a, b, c) \) are the direction ratios. Thus:
\[ l = \frac{-3}{\sqrt{26}}, \quad m = \frac{1}{\sqrt{26}}, \quad n = \frac{4}{\sqrt{26}} \]

### Step 5: Write the final answer
The direction cosines of the line joining the points \( (1, 2, -3) \) and \( (-2, 3, 1) \) are:
\[ \left( \frac{-3}{\sqrt{26}}, \frac{1}{\sqrt{26}}, \frac{4}{\sqrt{26}} \right) \]