Odisha Board Solutions for Chapter: Three Dimensional Geometry, Exercise 1: EXERCISE - 13(a)

If $A, B, C, D$ are the points $(6, 3, 2), (3, 5, 7), (2, 3,-1)$ and $(3, 5,-3)$ respectively, then find the projection of $\overline{AB}$ on $\overline{CD}$.

If $A, B, C$ are the points $(1,4, 2),(-2, 1, 2)$ and $(2,-3, 4)$ respectively then the angles of the triangle $ABC$ is of the form of $\frac{\mathrm{\pi }}{a}$, then $a=$

Prove that measure of the angle between two main diagonals of a cube is ${\cos }^{-1}\left(\frac{1}{3}\right)$.

Prove that measure of the angle between the diagonal of a face and the diagonal of a cube is ${\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)$.
 

If the angle which a diagonal of a cube makes with one of its edges is ${\cos }^{-1}\left(\frac{1}{\sqrt{k}}\right)$, then the value of $k$ is

Find the angle between the lines whose d.cs $l, m, n$ are connected by the relation, $3l+m+ 5n = 0$ and $6mn-2nl+5lm=0$.

Show that the measures of the angles between the four diagonals of a rectangular parallelepiped whose edges are $a, b, c$ are

${\cos }^{-1}\left(\frac{a^2\pm b^2\pm c^2}{a^2+b^2+c^2}\right)$.

If $l_1, m_1, n_1,$ and $l_2, m_2, n_2,$ are the direction cosines of two mutually perpendicular lines show that the direction cosines of the line perpendicular to both of them are $m_1{n}_2-m_2{n}_1, n_1{l}_2-n_2{l}_1, l_1{m}_2- l_2{m}_1$.