The direction cosines of the line drawn from `P(-5,3,1)` to Q(1,5,-2) is

Options

A. `(6,-2,-3)`
B. `(2,-4,1)`
C. `(-4,8,-1)`
D. `((6)/(7),(2)/(7),-(3)/(7))`

Answer

To find the direction cosines of the line drawn from point \( P(-5, 3, 1) \) to point \( Q(1, 5, -2) \), we will follow these steps:

### Step 1: Determine the direction ratios
The direction ratios of the line can be found by subtracting the coordinates of point \( P \) from the coordinates of point \( Q \).

\[ \text{Direction ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]

Where:
- \( P(x_1, y_1, z_1) = (-5, 3, 1) \)
- \( Q(x_2, y_2, z_2) = (1, 5, -2) \)

Calculating the direction ratios:
\[ \text{Direction ratios} = (1 - (-5), 5 - 3, -2 - 1) = (1 + 5, 2, -3) = (6, 2, -3) \]

### Step 2: Calculate the magnitude of the direction ratios
The magnitude of the direction ratios can be calculated using the formula:
\[ \text{Magnitude} = \sqrt{(a^2 + b^2 + c^2)} \]
Where \( a, b, c \) are the direction ratios.

Substituting the values:
\[ \text{Magnitude} = \sqrt{(6^2 + 2^2 + (-3)^2)} = \sqrt{(36 + 4 + 9)} = \sqrt{49} = 7 \]

### Step 3: Find the direction cosines
The direction cosines \( l, m, n \) can be found by dividing each direction ratio by the magnitude.

\[ l = \frac{a}{\text{Magnitude}}, \quad m = \frac{b}{\text{Magnitude}}, \quad n = \frac{c}{\text{Magnitude}} \]

Substituting the values:
\[ l = \frac{6}{7}, \quad m = \frac{2}{7}, \quad n = \frac{-3}{7} \]

### Final Result
The direction cosines of the line drawn from \( P(-5, 3, 1) \) to \( Q(1, 5, -2) \) are:
\[ \left( \frac{6}{7}, \frac{2}{7}, \frac{-3}{7} \right) \]

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