Find the direction ratios and the direction cosines of the line joining the points $(5, 4, -3)$ and $(3, -2, 6)$, the angle between the line and $x$-axis taken acute.

Find the angle between the lines whose direction cosines are $<\frac{\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}>$ and $<\frac{-\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}>$

Find the angle between the lines whose direction numbers are $< 1, 2, 1 >, < 2, - 3, 4 >$

Find the angle between the lines whose direction ratios are $<1, 1, 2>$and $<\sqrt{3}-1, -\sqrt{3}-1, 4>$.

Show that the lines with direction cosines $<\frac{12}{13}, -\frac{3}{13}, -\frac{4}{13}>, <\frac{4}{13}, \frac{12}{13}, \frac{3}{13}>$and$<\frac{3}{13}, -\frac{4}{13}, \frac{12}{13}>$ are mutually perpendicular.

Show that the points $A (4, 7, 8), B (2, 3, 4), C (- 1, - 2, 1) and D (1, 2, 5),$ taken in order, are the vertices of a parallelogram.