Two struts of a roof truss lie along lines with direction ratios (k, 2, -3) and (1, k, 4)…

Step 1. Perpendicular. $k \times 1 + 2 \times k + (-3) \times 4 = 0$ gives $3k = 12$, so $k = 4$.

Step 2. The length. The first strut has direction $(4, 2, -3)$. Its length is $\sqrt{16 + 4 + 9} = \sqrt{29} \approx 5.385$.

Step 3. The angle with the $x$-axis. $\cos\alpha = \dfrac{4}{\sqrt{29}} \approx 0.743$, so $\alpha \approx 42.03^\circ$.