Angle between a Line and a Plane

Find the distance of the point  $\left(-1, -5, -10\right)$, from the point of intersection of the line $\overrightarrow{r}=\left(2\widehat{i}-\widehat{j}+2\widehat{k}\right)+\lambda \left(3\widehat{i}+4\widehat{j}+2\widehat{k}\right)$ and the plane $\overrightarrow{r}\cdot \left(\widehat{i}-\widehat{j}+\widehat{k}\right)=5$.

The coordinate of the point where the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z+3}{4}$ meets the plane $x+y+4z=6$ is

Consider the plane $P:6x+4y+3z=12$, which intersects the coordinate axes at $A,B,C$. Locus of a point $Q$, such that volume of tetrahedron formed by points $Q,A,B,C$ is $4$ cubic units and a pair of planes $P_1$ and $P_2$ parallel to Plane $P$. Let a line through $\left(-1,3,2\right)$ with Direction ratios $-1,0,1$ be drawn to intersect $P_1$ and $P_2$ at $D \& D^{\text{'}}$ then which of the following is NOT true?

The perpendicular distance from the point $\left(3,1,1\right)$ on the plane passing through the point $\left(1,2,3\right)$ and containing the line, $\overrightarrow{r}=\hat{i}+\hat{j}+\lambda \left(2\hat{i}+\hat{j}+4\hat{k}\right)$ is

If the line $\frac{x-1}{4}=\frac{y+3}{-2}=\frac{z+5}{1}$ lies in the plane $2x+ly+mz=16,$ then $l^2+m^2$ is equal to

If planes $x-cy-bz=0, cx-y+az=0$ and $bx+ay-z=0$ pass through a straight line then $a^2+b^2+c^2=$

The measure of the angle between the line $\overrightarrow{r}=\left(2,-3,1\right)+k\left(2,2,1\right) ;k\in R$ and the plane $2x-2y+z+7=0$ is $\dots \dots \dots ..$

The perpendicular distance from the point of intersection of the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z}{-1}$ and plane $2x-y+z=0$ to the $Z-$axis is $\dots \dots \dots .$

If $\beta$ be the angle between the line $\overrightarrow{r}=\left(-\hat{i}+\hat{j}+2\hat{k}\right)+\lambda \left(\hat{i}+2\hat{j}+2\hat{k}\right)$ and the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+a\hat{k}\right)+4=0$ such that $\sin \beta =\frac{1}{3}$, then the value of $a$ can be

If the plane $2x – 3y – 6z = 13$ makes an angle ${\sin }^{-1}\left(\lambda \right)$ with the $x -$ axis, then the value of $\lambda$ is 

The angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{ȷ}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}·(2\hat{i}-\hat{ȷ}+\hat{k})=5$ is 

The angle between the plane $\overrightarrow{r}.\left(\hat{i}-2\hat{j}+3\hat{k}\right)=5$ and the line $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(\hat{i}-\hat{j}+\hat{k}\right)$ is 

The angle between the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}$ and the plane $2x+y-3z+4=0$ is 

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3 y-\alpha z+\beta =0$, then $(\alpha ,\beta )$ is

If the line $\overrightarrow{r}=(\hat{i}-2\hat{j}+3\hat{k})+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ is parallel to the plane $\overrightarrow{r}·(3\hat{i}-2\hat{j}+m\hat{k})=10$, then value of $m$ is
 

The angle between the line $\overrightarrow{r}=\left(i+\hat{\mathit{j}}-\hat{\mathit{k}}\right)+\lambda \left(3\hat{i}+\hat{\mathit{j}}\right)$ and the plane $\overrightarrow{r}·\left(\hat{\mathit{ı}}+2\hat{\mathit{ȷ}}+3\hat{\mathit{k}}\right)=8$ is

If the plane $ax-by+cz=0$ contains the line $\frac{x-a}{a}=\frac{y-2d}{b}=\frac{z-c}{c}, (b\ne 0)$, then $\frac{b}{d}$ is equal to

The line $\frac{x-1}{\lambda }=\frac{y}{2}=\frac{z+3}{-12}$ makes an isosceles triangle with the planes $x+2y-3z+5=0$ and $2x+3y-z+2=0,$ then value of $\lambda$ can be

If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point $P$ and the plane $3x+y+4z=16$ at a point $Q$, then $PQ$ is equal to