Direction cosines of the line passing through A(2,3, -1) and B(-3, 4, 2) are

Options

A. `(-5)/(sqrt35),(1)/(sqrt35),(3)/(sqrt35)`
B. `(5)/(sqrt35),(1)/(sqrt35),(4)/(sqrt35)`
C. `(-7)/(sqrt83),(3)/(sqrt83),(-5)/(sqrt83)`
D. `(-5)/(sqrt83),(-7)/(sqrt83),(-3)/(sqrt83)`

Answer

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

### Step 1: Find the direction ratios of the line
The direction ratios of the line can be found using the coordinates of points A and B. The direction ratios (dx, dy, dz) can be calculated as follows:

\[ dx = x_2 - x_1 = -3 - 2 = -5 \]
\[ dy = y_2 - y_1 = 4 - 3 = 1 \]
\[ dz = z_2 - z_1 = 2 - (-1) = 3 \]

So, the direction ratios of the line are (-5, 1, 3).

### Step 2: Calculate the magnitude of the direction ratios
The magnitude (length) of the direction ratios can be calculated using the formula:

\[ \text{Magnitude} = \sqrt{dx^2 + dy^2 + dz^2} \]

Substituting the values we found:

\[ \text{Magnitude} = \sqrt{(-5)^2 + (1)^2 + (3)^2} = \sqrt{25 + 1 + 9} = \sqrt{35} \]

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

\[ l = \frac{dx}{\text{Magnitude}} = \frac{-5}{\sqrt{35}} \]
\[ m = \frac{dy}{\text{Magnitude}} = \frac{1}{\sqrt{35}} \]
\[ n = \frac{dz}{\text{Magnitude}} = \frac{3}{\sqrt{35}} \]

Thus, the direction cosines of the line are:

\[ \left( \frac{-5}{\sqrt{35}}, \frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}} \right) \]

### Final Answer:
The direction cosines of the line passing through points A(2, 3, -1) and B(-3, 4, 2) are:

\[ \left( \frac{-5}{\sqrt{35}}, \frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}} \right) \]

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