Odisha Board Solutions for Chapter: Three Dimensional Geometry, Exercise 4: ADDITIONAL EXERCISES

The angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)=4$ is ${\sin }^{-1}\left(\frac{k\sqrt{2}}{3}\right)$. Find the value of $k$.

Prove that the acute angle between the lines whose direction cosines are given by the relations $l+m+n=0$ and $l^2+m^2-n^2=0$ is $\frac{\mathrm{\pi }}{3}$

Prove that the three lines drawn from origin with direction cosines $l_1, m_1, n_1; l_2, m_2, n_2; l_3, m_3, n_3$, are coplanar if $\left|\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right|=0$.

Prove that three lines drawn from origin with direction cosines proportional to $(1,-1, 1), (2-3,0), (1,0,3)$ lie on one plane.

Determine $k$ so that the lines joining the points $P_1(k, 1,-1)$ and $P_2(2k,0,2)$ shall be perpendicular to the line from $P_2$ to $P_3(2+2k,k,1)$.

The angle between the lines whose direction ratios are proportional to $a,b,c$ and $b-c, c-a, a-b$ is of the form of $\frac{\mathrm{\pi }}{a}$, then $a=$

$\mathrm{O}$ is the origin and $\mathrm{A}$ is the point $(a,b,c)$. Find the equation of the plane through $\mathrm{A}$ at right angles to $\mathrm{OA}$.

Find the equation of the plane through $(6,3,1)$ and $(8,-5,3)$ parallel to $x-$axis.