Find the direction-cosines of the line joining the points (-2,4,-5) and (1,2,3).

Answer

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

### Step 1: Identify the Points
Let the points be:
- Point P = (-2, 4, -5)
- Point Q = (1, 2, 3)

### Step 2: Find the Direction Ratios
The direction ratios of the line joining points P and Q can be found using the formula:
\[ \text{Direction Ratios} = Q - P \]
Calculating each component:
- For x-coordinate: \(1 - (-2) = 1 + 2 = 3\)
- For y-coordinate: \(2 - 4 = -2\)
- For z-coordinate: \(3 - (-5) = 3 + 5 = 8\)

Thus, the direction ratios are:
\[ (3, -2, 8) \]

### Step 3: Calculate the Modulus of the Direction Ratios
The modulus (length) of the direction ratios can be calculated using the formula:
\[ \text{Modulus} = \sqrt{(3^2) + (-2^2) + (8^2)} \]
Calculating each term:
- \(3^2 = 9\)
- \((-2)^2 = 4\)
- \(8^2 = 64\)

Now, summing these:
\[ \text{Modulus} = \sqrt{9 + 4 + 64} = \sqrt{77} \]

### Step 4: Find the Direction Cosines
The direction cosines (l, m, n) can be found by dividing each direction ratio by the modulus:
\[ l = \frac{3}{\sqrt{77}}, \quad m = \frac{-2}{\sqrt{77}}, \quad n = \frac{8}{\sqrt{77}} \]

### Final Result
Thus, the direction cosines of the line joining the points (-2, 4, -5) and (1, 2, 3) are:
\[ \left( \frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}} \right) \]

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