If the lines $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{λ}$ and $\frac{x}{1}=\frac{y}{2λ}=\frac{z}{3}$ are perpendicular to each other, then $λ$ is equal to

1

The correct answer is Option (1) → 1

Let the direction ratios of the first line be:

$a_1 = 7, \quad b_1 = -5, \quad c_1 = \lambda$

Let the direction ratios of the second line be:

$a_2 = 1, \quad b_2 = 2\lambda, \quad c_2 = 3$

If the lines are perpendicular, then their dot product is zero:

$a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$

$7 \cdot 1 + (-5) \cdot (2\lambda) + \lambda \cdot 3 = 0$

$7 - 10\lambda + 3\lambda = 0$

$7 - 7\lambda = 0$

$\lambda = 1$