Question

Find the angle between the lines \[ \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{-5} \quad \text{and} \quad \frac{x+3}{-3} = \frac{y-1}{2} = \frac{z-5}{5}. \] 

Hint:

The formula for the angle between two lines can be simplified by using the dot product of their direction ratios.

Answer 1

Step 1: The direction ratios of the first line are \( (-1, 2, -5) \) and for the second line, they are \( (-3, 2, 5) \). 

Step 2: The angle \( \theta \) between the two lines is given by the formula: \[ \cos \theta = \frac{l_1 l_2}{|l_1| |l_2|} \] where \( l_1 \) and \( l_2 \) are the direction ratios of the two lines. 

Step 3: Substitute the values of \( l_1 = (-1, 2, -5) \) and \( l_2 = (-3, 2, 5) \) into the formula and calculate: \[ \cos \theta = \frac{(-1)(-3) + (2)(2) + (-5)(5)}{\sqrt{(-1)^2 + 2^2 + (-5)^2} \times \sqrt{(-3)^2 + 2^2 + 5^2}}. \] 

Step 4: Simplify the equation and solve for \( \theta \).