The angle between the lines (2x-1)/4=(y+3)/2=(1-z)/1 and (x+2)/1=(2y-…

The correct answer is Option (2) → \(\displaystyle\frac{\pi}{4}\)

Each line must first be put in the standard form, with coefficient +1 for x, y and z.

First line:

    \( \displaystyle \frac{2x-1}{4} = \frac{x-\frac{1}{2}}{2} \quad\text{and}\quad \frac{1-z}{1} = \frac{z-1}{-1} \)

so its direction ratios are 2, 2, −1.

Second line:

    \( \displaystyle \frac{2y-3}{8} = \frac{y-\frac{3}{2}}{4} \)

so its direction ratios are 1, 4, 1.

The angle between the lines:

    \((2)(1)+(2)(4)+(-1)(1) = 9\)

    \(\sqrt{4+4+1} = 3\) and \(\sqrt{1+16+1} = 3\sqrt{2}\)

    \( \displaystyle \cos\theta = \frac{9}{3 \times 3\sqrt{2}} = \frac{1}{\sqrt{2}} \)

    \( \displaystyle \theta = \frac{\pi}{4} \)