The angle between two lines \(\frac{x+1}{2}=\frac{y+3}{2}=\frac{z-4}{-1} \) and \( \frac{x-4}{1}=\frac{y+4}{2}=\frac{z+1}{2}\) is:

Calculation

The angle θ between the two lines

\(\frac{x-x_1}{a_1} = \frac{y-y_1}{a_2} = \frac{z-z_1}{a_3}\)

and \(\frac{x-x_2}{b_1} = \frac{y-y_2}{b_2} = \frac{z-z_2}{b_3}\) is given by:

cos θ = \(\frac{a_1b_1 + a_2b_2 + a_3b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2}\sqrt{b_1^2 + b_2^2 + b_3^2}}\)

Now in the given equation:

a₁ = 2, a₂ = 2, a₃ = -1

b₁ = 1, b₂ = 2, b₃ = 2

∴ cos θ = \(\frac{2 \times 1 + 2 \times 2 + (-1) \times 2}{\sqrt{4+4+1}\sqrt{4+4+1}} = \frac{4}{9}\)

⇒ θ = cos^-1(\(\frac{4}{9}\))

Hence option 2 is correct