Find the angle between the two lines :
`(x+1)/(2)=(y-2)/(5)=(z+3)/(4)and(x-1)/(5)=(y+2)/(2)=(z-1)/(-5)`

Answer

To find the angle between the two lines given by the equations:

1. \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}\)
2. \(\frac{x-1}{5} = \frac{y+2}{2} = \frac{z-1}{-5}\)

we will follow these steps:

### Step 1: Identify the Direction Ratios

From the first line, we can extract the direction ratios:
- For the first line, the direction ratios \(a_1, b_1, c_1\) are \(2, 5, 4\) respectively.

From the second line, we can extract the direction ratios:
- For the second line, the direction ratios \(a_2, b_2, c_2\) are \(5, 2, -5\) respectively.

### Step 2: Use the Formula for the Angle Between Two Lines

The formula for the cosine of the angle \(\theta\) between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is given by:

\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Substitute the Direction Ratios into the Formula

Substituting the values we found:

- \(a_1 = 2\), \(b_1 = 5\), \(c_1 = 4\)
- \(a_2 = 5\), \(b_2 = 2\), \(c_2 = -5\)

Now, calculate the numerator:

\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = (2)(5) + (5)(2) + (4)(-5) = 10 + 10 - 20 = 0 \]

Now, calculate the denominator:

\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + 5^2 + 4^2} = \sqrt{4 + 25 + 16} = \sqrt{45} \]

\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{5^2 + 2^2 + (-5)^2} = \sqrt{25 + 4 + 25} = \sqrt{54} \]

### Step 4: Calculate \(\cos \theta\)

Now substituting back into the formula:

\[ \cos \theta = \frac{0}{\sqrt{45} \cdot \sqrt{54}} = 0 \]

### Step 5: Determine the Angle \(\theta\)

Since \(\cos \theta = 0\), this means:

\[ \theta = \frac{\pi}{2} \text{ or } 90^\circ \]

### Conclusion

The angle between the two lines is \(90^\circ\), indicating that the lines are perpendicular to each other.

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