The angle between the line 2x = 3y = z and x-axis is:

The correct answer is Option (2) → \(\cos^{-1}\left(\displaystyle\frac{3}{7}\right)\)

Writing the line in symmetric form:

    \( \displaystyle \frac{x}{1/2} = \frac{y}{1/3} = \frac{z}{1} \)

The direction ratios are \(\frac{1}{2}, \frac{1}{3}, 1\). Multiplying by 6, they are 3, 2, 6, with

    \(\sqrt{3^2 + 2^2 + 6^2} = \sqrt{49} = 7\)

So the direction cosines are \(\frac{3}{7}, \frac{2}{7}, \frac{6}{7}\). The x-axis has direction \(\hat{\imath}\), so the angle with it is given by the first direction cosine:

    \( \displaystyle \cos\theta = \frac{3}{7},\quad \theta = \cos^{-1}\left(\frac{3}{7}\right) \)