Three Dimension...

If  $$A(3\hat { i } +2\hat { j } +3\hat { k } ),B(-\hat { i } -\hat { j } +8\hat { k } ),C(-4\hat { i } +4\hat { j } +6\hat { k } )$$  are the vertices of a triangle then the equation of the line passing through the circumcentre and parallel to  $$\vec { A B }$$  is

The cartesian from of equation a line passing through the point position vector $$2\hat{i}-\hat{j}+2\hat{k}$$ and is in the direction of $$-2\hat{i}+\hat{j}+\hat{k}$$, is

If $$\cos { \alpha ,\quad \cos { \beta ,\quad \cos { \gamma  }  }  }$$   are the direction cosine of a line, then find the value of $${ cos }^{ 2 }\alpha +\left( \cos { \beta +\sin { \gamma  }  }  \right)$$$$\left( \cos { \beta - \sin { \gamma  } }  \right)$$

$$\dfrac { x - 2 } { 1 } = \dfrac { y - 3 } { 1 } = \dfrac { z - 4 } { - 1 }$$ & $$\dfrac { x - 1 } { k } = \dfrac { y - 4 } { 2 } = \dfrac { z - 5 } { 2 }$$ are coplanar then k=?

A normal to the plane $$  x=2  $$ is...

The straight lines $$ \dfrac {x-1}{1} = \dfrac {y-2}{2} = \dfrac {z-3}{3} $$ and $$ \dfrac {x-1}1{} = \dfrac {y-2}{2} = \dfrac {z-3}3{} $$ are :

Direction ratio of line given by $$\dfrac { x-1 }{ 3 } =\dfrac { 6-2y }{ 10 } =\dfrac { 1-z }{ -7 } $$ are:

A line with direction cosines proportional to 2 , 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by ______________.

the points $$(\alpha ,\beta )(\gamma ,\delta ),(\alpha ,\delta )and (\gamma ,\beta )$$   where $$\alpha ,\beta ,\gamma ,\delta $$  are different real numbers, are 

The angle between the lines whose direction cosines are given by $$2l-m+2n=0$$, $$lm+mn+nl=0$$ is