Interaction between Two Planes

Let the equation of the plane passing through the points $\left(1,2,6\right), \left(2,5,7\right)$ and perpendicular to the plane $4x-y+z=15$ be $ax+by+cz+d=0$. If $a, b, c, d\in Z$ with gcd $\left(\left|a\right|,\left|b\right|,\left|c\right|\right)=1$ then $\left|a+b+c+d\right|$ is equal to

The distance between the planes $\overrightarrow{\mathit{r}}·\left(2\hat{i}+\hat{j}-2\hat{k}\right)+5=0$ and $\overrightarrow{r}·\left(6\hat{i}+3\hat{j}-6\hat{k}\right)+2=0$ is

Value of $k$, for which the planes $x+2y+kz=0$ and $2x+y-2z=0$ are at right angles, is

The equations $xy=0$ in three dimensional space is represented by :

An equation of a plane parallel to the plane $x-2y+2z-5=0$ and at a unit distance from the origin is

If the angle between the planes $\overrightarrow{r}.\left(m \widehat{i}- \widehat{j}+2 \widehat{k}\right)+3=0$and$\overrightarrow{r}.(2 \widehat{i}- m\widehat{j}-\widehat{k})-5=0$ is $\frac{\pi }{3}$ then $m=$

The equation of the plane passes through $\text{P}\left(2,\mathrm{ }3,\mathrm{ }4\right)$ and parallel to the plane $\mathrm{x}+2\mathrm{y}+4\mathrm{z}=5$ is -

If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0$ , respectively, then the maximum value of $\lambda +\mu$ is equal to:

If the equation of the plane passing through $(1,1,1)$ and parallel to the plane $x+2y+3z-7=0$ is of the form $x+2y+3z=k$ then find the value of $k$

If $\theta$ is the angle between the planes $x+2y+2z-5=0$ and $3x+3y+2z-8=0$ then $\theta =$

If $\theta$ is the acute angle between the planes $x+y-z=4 and x+2 y+z=9$ then, $\theta =$

The acute angle between the planes $2x-y+z=5\text{ and }x+y+2z=7$ is

The plane $\overrightarrow{r}·\left(4\hat{i}+7\hat{j}+4\hat{k}\right)=-81$ is rotated through a right angle about its line of intersection with the plane $\overrightarrow{r}·\left(5\hat{i}+3\hat{j}+10\hat{k}\right)=25$. The equation of the plane in its new position is

The distance between the parallel planes $x+2y-2z+4=0$ and $x+2y-2z-8=0$ is 

The distance between the parallel planes $2x+3y+4z=4$ and $4x+6y+8z=12$ is 

Find the value of $\lambda$ for which the planes $x-4y+\lambda z+3=0\text{ and }2x+2y+3z=5$ are perpendicular to each other.

If the angle between the planes $\overrightarrow{r}·(\hat{i}+\hat{j}-2\hat{k})=3$ and $2x-2y+z = 2$ is ${\cos }^{-1}\left(\frac{a}{b\sqrt{c}}\right)$, where $a,b,c\in \mathrm{I}$ then $a+b+c=$

Find the distance between the parallel planes $x+y-z+4=0$ and $x+y-z+5=0$. [Write the answer without units in the answer box].

Find the value of $p$, if the angle between the planes $\overrightarrow{r}·\left(p\hat{i}-\hat{j}+2\hat{k}\right)+3=0$ and $\overrightarrow{r}.(2\hat{i}-p\hat{j}-\hat{k})-5=0$ is $\frac{\pi }{3}$.