Find the position vector of a point $A$ in space such that $\overrightarrow{\mathbf{O A}}$ is inclined at $60^{\circ}$ to $O X$ and at $45^{\circ}$ to $O Y$ and $|\overrightarrow{\mathbf{O A}}|=10$ units.

Find the vector equation of the line which is parallel to the vector $3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ and which passes through the point $(1,-2,3)$.

Show that the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ intersect. Also, find their point of intersection.

Find the angle between the lines $$ \overrightarrow{\mathbf{r}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \overrightarrow{\mathbf{r}}=(2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}})+\mu(6 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}) . $$

Prove that the line through $A(0,-1,-1)$ and $B(4,5,1)$ intersects the line through $C(3,9,4)$ and $D(-4,4,4)$.

Prove that the lines $x=p y+q, z=r y+s$ and $x=p^{\prime} y+q^{\prime}, z=r^{\prime} y+s^{\prime}$ are perpendicular, if $p p^{\prime}+r r^{\prime}+1=0$.

Find the equation of a plane which bisects perpendicularly the line joining the points $A(2,3,4)$ and $B(4,5,8)$ at right angles.

Find the equation of a plane which is at a distance $3 \sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axis.

If the line drawn from the point $(-2,-1,-3)$ meets a plane at right angle at the point $(1,-3,3)$, then find the equation of the plane.

Find the equation of the plane through the points $(2,1,0),(3,-2,-2)$ and $(3,1,7)$.

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi}{3}$ each.

Find the angle between the lines whose direction cosines are given by the equation $l+m+n=0$ and $l^2+m^2-n^2=0$.

If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l+\delta l, m+\delta m, n+\delta n$, then show that the small angle $\delta \theta$ between the two positions is given by $\delta \theta^2=\delta l^2+\delta m^2+\delta n^2$.

If 0 is the origin and $A$ is $(a, b, c)$, then find the direction cosines of the line $O A$ and the equation of plane through $A$ at right angle to $O A$.

Two systems of rectangular axis have the same origin. If a plane cuts them at distances $a, b, c$ and $a^{\prime}, b^{\prime}, c^{\prime}$, respectively from the origin, then prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$.

Find the foot of perpendicular from the point $(2,3,-8)$ to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$. Also, find the perpendicular distance from the given point to the line.

Find the distance of a point $(2,4,-1)$ from the line $$\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}.$$

Find the length and the foot of perpendicular from the point $\left(1, \frac{3}{2}, 2\right)$ to the plane $2 x-2 y+4 z+5=0$.

Find the equation of the line passing through the point $(3,0,1)$ and parallel to the planes $x+2 y=0$ and $3 y-z=0$.

Find the equation of the plane through the points $(2,1,-1),(-1,3,4)$ and perpendicular to the plane $x-2 y+4 z=10$.

Find the shortest distance between the lines gives by $\begin{array}{ll} & \overrightarrow{\mathbf{r}}=(8+3 \lambda) \hat{\mathbf{i}}-(9+16 \lambda) \hat{\mathbf{j}}+(10+7 \lambda) \hat{\mathbf{k}} \\ \text { and } & \overrightarrow{\mathbf{r}}=15 \hat{\mathbf{i}}+29 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}+\mu(3 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}) .\end{array}$

Find the equation of the plane which is perpendicular to the plane $5 x+3 y+6 z+8=0$ and which contains the line of intersection of the planes $x+2 y+3 z-4=0$ and $2 x+y-z+5=0$.

If the plane $a x+b y=0$ is rotated about its line of intersection with the plane $z=0$ through an angle $\alpha$, then prove that the equation of the plane in its new position is $a x+b y \pm\left(\sqrt{a^2+b^2} \tan \alpha\right) z=0$.

Find the equation of the plane through the intersection of the planes $\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+3 \hat{\mathbf{j}})-6=0$ and $\overrightarrow{\mathbf{r}} \cdot(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-4 \hat{\mathbf{k}})=0$, whose perpendicular distance from origin is unity.

Show that the points $(\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ and $3(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ are equidistant from the plane $\overrightarrow{\mathbf{r}} \cdot(5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-7 \hat{\mathbf{k}})+9=0$ and lies on opposite side of it.

$\overrightarrow{\mathbf{A B}}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{C D}}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ are two vectors. The position vectors of the points $A$ and $C$ are $6 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $-9 \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$, respectively. Find the position vector of a point $P$ on the line $A B$ and a point $Q$ on the line $C D$ such that $\overrightarrow{\mathbf{P Q}}$ is perpendicular to $\overrightarrow{\mathbf{A B}}$ and $\overrightarrow{\mathbf{C D}}$ both.

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

If $l_1, m_1, n_1, l_2, m_2, n_2$ and $l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines, then prove that the line whose direction cosines are proportional to $l_1+l_2+l_3, m_1+m_2+m_3$ and $n_1+n_2+n_3$ makes equal angles with them.

Distance of the point $(\alpha, \beta, \gamma)$ from $Y$-axis is

If the direction cosines of a line are $k, k$ and $k$, then

The distance of the plane $\overrightarrow{\mathbf{r}}\left(\frac{2}{7} \hat{\mathbf{i}}+\frac{3}{7} \hat{\mathbf{j}}-\frac{6}{7} \hat{\mathbf{k}}\right)=1$ from the origin is

The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ and the plane $2 x-2 y+z=5$ is

The reflection of the point $(\alpha, \beta, \gamma)$ in the $X Y$-plane is

The area of the quadrilateral $A B C D$ where $A(0,4,1)$, $B(2,3,-1), C(4,5,0)$, and $D(2,6,2)$ is equal to

The locus represented by $x y+y z=0$ is

If the plane $2 x-3 y+6 z-11=0$ makes an angle $\sin ^{-1} \alpha$ with $X$-axis, then the value of $\alpha$ is

If a plane passes through the points $(2,0,0)(0,3,0)$ and $(0,0,4)$ the equation of plane is ................ .

The direction cosines of the vector $(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})$ are .............. .

The vector equation of the line $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$ is ............ .

The vector equation of the line through the points $(3,4,-7)$ and $(1,-1$, 6) is ................. .

The cartesian equation of the plane $\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})=2$ is ............. .

The unit vector normal to the plane $x+2 y+3 z-6=0$ is $$\frac{1}{\sqrt{14}} \hat{\mathbf{i}}+\frac{2}{\sqrt{14}} \hat{\mathbf{j}}+\frac{3}{\sqrt{14}} \hat{\mathbf{k}}.$$

The intercepts made by the plane $2 x-3 y+5 z+4=0$ on the coordinate axis are $-2, \frac{4}{3}$ and $-\frac{4}{5}$.

The angle between the line $\overrightarrow{\mathbf{r}}=(5 \hat{\mathbf{i}}-\hat{\mathbf{j}}-4 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})$ and the plane $\overrightarrow{\mathbf{r}}(3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-\hat{\mathbf{k}})+5=0$ is $\sin ^{-1}\left(\frac{5}{2 \sqrt{91}}\right)$.

The angle between the planes $\overrightarrow{\mathbf{r}}(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})=1$ and $\overrightarrow{\mathbf{r}}(\hat{\mathbf{i}}-\hat{\mathbf{j}})=4$ is $$\cos ^{-1}\left(\frac{-5}{\sqrt{58}}\right)$$

The line $\overrightarrow{\mathbf{r}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ lies in the plane $\overrightarrow{\mathbf{r}}(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+2=0$.

The vector equation of the line $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$ is $\overrightarrow{\mathbf{r}}=5 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}+\lambda(3 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$

The equation of a line, which is parallel to $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and which passes through the point $(5,-2,4)$ is $\frac{x-5}{2}=\frac{y+2}{-1}=\frac{z-4}{3}$.

If the foot of perpendicular drawn from the origin to a plane is $(5,-3,-2)$, then the equation of plane is $\overrightarrow{\mathbf{r}}(5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})=38$.