Find the direction cosines of a line which is perpendicular to the two lines formed by joining $A (2, 3, -4) \mathrm{to} B (-3, 3, - 2) \mathrm{and} C (- 1, 4, 2) \mathrm{to} D (3, 5, 1).$

Show that the straight lines whose direction cosines are given by $2l+ 2m-n = 0$ and $mn+nl+lm= 0$ are at right angles.

Find the angle between the lines whose direction cosines are given by the equations $3l+m+5n = 0$ and $6mn-2nl+5lm = 0$.

Prove that the lines whose direction cosines are given by $u{l}^2+u{m}^2+ w{n}^2 = 0$ and $al+bm+cn=0$ are perpendicular if$u(b^2+c^2)+v(c^2+a^2)+w(a^2+b^2)=0$.

Prove that the lines whose direction cosines are given by $u{l}^2+u{m}^2+ w{n}^2 = 0$ and $al+bm+cn=0$ are parallel if $\frac{a^2}{u}+\frac{b^2}{v}+\frac{c^2}{w}=0$.

The fig. shows a rectangular box of size a, b and c units along x, y and z-axes respectively. Prove that the angles between the four diagonals are given by ${\cos }^{-1}\left(\frac{\pm a^2\pm b^2\pm c^2}{a^2+b^2+c^2}\right)$

where all signs cannot be +ve or -ve.