Three Dimension...

The Equation of the plane through a point $$\displaystyle  2\hat{i}-\hat{j}+4\hat{k} $$ & parallel to the plane $$\displaystyle \overline{r}.\left ( 2\hat{i}+4\hat{j}-7\hat{k} \right )=6$$ is

If the foot of the perpendicular from the origin to a plane is $$\displaystyle \left ( a,b,c \right )$$, the equation of the plane is

Let $$N$$ be the foot of the perpendicular of length $$p$$, from the origin to a plane and $$l$$, $$m$$, $$n$$ be the direction cosines of $$ON$$, the equation of the plane is

The scalar product form of equation of plane $$\displaystyle \overline{r}=\left ( s-2t \right )\hat{i}+\left ( 3-t \right )\hat{j}+\left ( 2s-t \right )\hat{k}$$ is

For what value of $$m$$, the points $$(3,5)$$, $$(m,6)$$ and $$\begin{pmatrix} \dfrac { 1 }{ 2 },\dfrac {15 }{ 2 } \end{pmatrix}$$ are collinear?

If $${ l }_{ 1 },{ m }_{ 1 },{ n }_{ 1 }$$ and $${ l }_{ 2 },{ m }_{ 2 },{ n }_{ 2 }$$ are DCs of the two lines inclined to each other at an angle $$\theta$$, then the DCs of the internal bisector of the angle between these lines are

If the projection of a line segment on $$x,y$$ and $$z$$ axes are respectively $$3,4$$ and $$5$$, then the length of the line segment is

Find the value of $$p$$ for which the points $$(-5,1)$$, $$(1,p)$$ and $$(4, -2)$$ are collinear.

The position vectors of point $$A$$ and $$B$$ are $$\hat { i } -\hat { j } +3\hat { k } $$ and $$3\hat { i } +3\hat { j } +3\hat { k } $$ respectively. The equation of a plane is $$r.\left( 5\hat { i } +2\hat { j } -7\hat { k }  \right) +9=0$$. The point $$A$$ and $$B$$

Lines $$OA,OB$$ are drawn from $$O$$ with direction cosines proportional to $$(1,-2,-1),(3,-2,3).$$ Find the direction cosines of the normal to the plane $$AOB$$