Angle Between a Line and a Plane

The coordinates of the point where the line  $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z+3}{4}$  meets the plane $x+y+4z=6$  would be:

Find the angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{ȷ}-k)+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}\left(2\stackrel{⏜}{i}-\stackrel{⏜}{j}+\stackrel{⏜}{k}\right)=5$.

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

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

Consider the plane $x+y-z=1$ and point $A(1,2,-3)$ . A line $L$ has the equation $x=1+3r, y=2-r \& z=3+4r.$

The equation of the plane containing line $L$ and point $A$ has the equation

For next three question please follow the same 

Consider the plane $x+y-z=1$ and point $A(1,2,-3)$ . A line $L$ has the equation $x=1+3r, y=2-r \& z=3+4r.$

 The coordinate of a point $B$ of line $L$ such that $AB$ is parallel to the plane is

Show that the line of intersection of the planes $\overrightarrow{r}·(\hat{i}+3\hat{j}-2\hat{k})=0$ and $\overrightarrow{r}\left(2\hat{i}+4\hat{j}-3\hat{k}\right)=0$ is equally inclined to $\hat{i}$ & $\hat{j}$. Also find the angle it makes with $\hat{k}$.

Find the value of $k$. If the  angle between the line $\overrightarrow{r}=(\hat{i}+2\hat{ȷ}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and the plane $\overrightarrow{r}\left(2\hat{i}-\hat{j}+\hat{k}\right)=5$ is ${\sin }^{-1}\left(\frac{2\sqrt{2}}{k}\right)$.

Prove that the lines $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-6}{7}$ and  $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ are coplanar. Also, find the equation of the plane containing these two lines.

Find the angle at which the normal vector to the plane $4x+8y+z=5$ is inclined to the coordinate axes.  

If the line $\overrightarrow{r}=\hat{i}+\mathrm{\lambda }\left(2\hat{i}-m\hat{j}-3\hat{k}\right)$ is parallel to the plane $\overrightarrow{r}·\left(m\hat{i}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=4$, then find the value of $m$.

Find the angle between the line $\overrightarrow{r}=\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)+\mathrm{\lambda }\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}\right)$ and plane $\stackrel{\rightharpoonup }{\mathrm{r}}·\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)=4$

Find the angle between the line $\overrightarrow{r}=\left(\hat{i}+2\hat{j}-\hat{k}\right)+\mathrm{\lambda }\left(\hat{i}-\hat{j}+\hat{k}\right)$ and plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)=4$

 Find the angle between the line $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-2}{2}$ and plane $3x+4y+z+5=0$.

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

Find the coordinates of the point where the line $\frac{\mathrm{x}+1}{2}=\frac{\mathrm{y}+2}{3}=\frac{\mathrm{z}+3}{4}$ meets the plane $\mathrm{x}+\mathrm{y}+4\mathrm{z}=6$.

Distance of point of $(-2,3,-4)$ from the line $\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}$ measured parallel to the plane $4 x+12 y-3 z+1=0$ is $\lambda$ then $\lambda$ is

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if :

The line passing through the points $\left(5, 1, a\right)$ and $\left(3, b, 1\right)$ crosses the $yz$-plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$. Then, find the value of $a$and $b$:

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