The line of intersection of the planes $\overrightarrow { r } .\left( 3\hat { i } -\hat { j } +\hat { k }  \right) =1$ and $\overrightarrow { r } .\left( \hat { i } +4\hat { j } -2\hat { k }  \right) =2$ is parallel to vector

There are two different planes, one passing though the x-axis and the other passing through y-axis. The angle between the planes is $\cfrac{\pi}{4}$. Then locus of a point on the line of intersection of the planes in.

The line of intersection of the planes 
$r.\left( {3\hat i - \hat j + \hat k} \right) = 1$ and $r.\left( {\hat i + 4\hat j - 2\hat k} \right) = 2$ is parallel to the vector

A unit vector parallel to the intersection of the planes $\vec r\cdot (\hat i-\hat j+\hat k)=5$ and $\vec r\cdot (2\hat i+\hat j-3\hat k)=4$ can be

Let L be the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$. If L makes an angle $\alpha$ with the positive x-axis, then $cos\alpha$ equals:

A non-zero vector $\vec{a}$ is parallel to the line of intersection of the plane determined  by the vectors $\hat{i},\hat{i}+\hat{j}$ and the plane determined by the vectors $\hat { i } -\hat { j } ,\hat { i } -\hat { k }$. The angle between $\vec{a}$ and $\hat { i } -2\hat { j } +2\hat { k } $ is

The planes $bx-ay=n,cy-bz=1,az-cx=m$ intersect in a line if

Let $L$ be the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$.

The equation of plane through the line of intersection of the planes $2x+3y+4z-7=0, x+y+z-1=0$ and perpendicular to the plane $x-5y+3z-6=0$ is

The direction cosines of a line parallel to the planes $\displaystyle 3x + 4y + z = 0$ and $\displaystyle x - 2y - 3z = 5$ are

If $\displaystyle \left ( 3, : \lambda, : \mu \right )$ is a point on the line then $\displaystyle 2x + y + z = 0 = x - 2y + z -1$ then

The variable plane $\displaystyle \left ( 2 \lambda + 1 \right )x + \left ( 3 - \lambda \right )y + z = 4$ always passes through the line 

The equation of the plane which contains the origin and the line of intersection of the planes $\vec r.\vec a=\vec p$ and $\vec r.\vec b=\vec q$ is

The distance of the point $(1, -2, 3)$ from the plane $x-y+z=5$ measured parallel to the line. $\frac { x }{ 2 } =\frac { y }{ 3 } =\frac { z }{ -6 } ,\quad is:$

Which of the following does not represent a straight line?

Consider a plane $x+2y+3z=15$ and a line $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{4}$ then find the distance of origin from point of intersection of line and plane.

Let $L$ be the line of intersection of the planes $2x+3y+z= 1$ and $x+3y+2z= 2$ . If $L$ makes an angle $\alpha $ with the positive $x$ -axis, then $\cos \alpha$ equals 

The vector equation of the line of intersection of the planes $r.(i+2j+3k)=0$ and $r.(3i+2j+k)=0$ is

The direction ratios of the line $x-y+z-5=0=x-3y-6$ are 

The line of intersection of the planes $\overrightarrow { r } .\left( 3i-j+k \right) =1$ and $\overrightarrow { r } .\left( i+4j-2k \right) =2$ is parallel to the vector:

Consider the planes  $3x - 6y - 2z = 15$  and  $2x + y - 2z = 5$.  Which of the following vectors is parallel to the line of intersection of given plane

The equations of the line of intersection of the planes $\displaystyle x + y + z = 2$ and $\displaystyle 3x - y + 2z = 5$ in symmetric form are

Consider the planes $\displaystyle 3x-6y-2z=15$ and $\displaystyle 2x+y-2z=5.$ 

The line of intersection of the planes $\displaystyle \bar r (3\hat i - \hat j + \hat k) = 1$ and $\displaystyle \bar r (\hat i + 4\hat j - 2\hat k) = 2$ is parallel to the vector

Consider three planes$P _1: x-y+z=1$$P _2: x+y-z=-1$$P _3: x-3y+3z=2$Let $L _1, L _2, L _3$ be the lines of intersection of the planes ${P} _{2}$ and ${P} _{3},\ {P} _{3}$ and ${P} _{1}$, and ${P} _{1}$ and ${P} _{2}$, respectively.
STATEMENT-$1$ : At least two of the lines ${L} _{1},\ {L} _{2}$ and ${L} _{3}$ are non-parallel.
and 
STATEMENT -$2$ : The three planes do not have a common point.

Let L be the line of intersection of the planes $2x + 3y + z = 1$ and $x + 3y + 2z = 2$. If L makes an angle $\alpha$ with the positive x-axis, then $\cos \alpha$ equals

Find the angle between the line of intersection of the planes $\overrightarrow { r } .\left( i+2j+3k \right) =0$ and $\overrightarrow { r } .\left( 3i+2j+3k \right) =0$ with coordinate axes