Vector Algebra ...

If $$\vec { x }$$ and $$\vec { y } $$ be unit vectors and $$\left| \vec { z }  \right| =\frac { 2 }{ \sqrt { 7 }  } $$ such that $$\vec { z } +\vec { z } \times \vec { x } =\vec { y } ,$$ then the angle $$\theta $$ between $$\vec { x }$$  and $$\vec { z } $$ can be 

Let $$a=(1,-2,3)$$ and $$b=(2,7,4)$$ then

Let $$\hat {a}$$ and $$\hat {b}$$ two unit vector such that $${ \left( \hat { a } .\hat { b }  \right)  }^{ 2 }-\left| \hat { a } \times \hat { b }  \right| $$ is maximum then $$\left| \hat { a } .\hat { b }  \right|$$ is equal to

The cartesian equation of the plane perpendicular to vector $$3\bar {i}-2\bar {j}-2\bar {k}$$ and passing through the point $$2\bar {i}+3\bar {j}-\bar {k}$$ is

The position vectors of two vertices and the centroid of a triangle are $$\overset { \rightarrow  }{ i } +\overset { \rightarrow  }{ j } ,\overset { \rightarrow  }{ 2i } -\overset { \rightarrow  }{ j } +\overset { \rightarrow  }{ k } $$ and $$\overset { \rightarrow  }{ k } $$ respectively. The position vector of the third vertex of the triangle is :

If $$u,\ v,\ w$$ are non-coplanar vector and $$p,\ q$$ are real numbers, then the equality $$[3u\ pv\ pw]-[pv\ w\ qw]-[2w\ qv\ qu]=0$$ holds for 

Unit vector perpendicular to the plane of the triangle  $$ABC$$  with position vectors of the vertices  $$A , B , C ,$$  is  $$($$ where  $$\Delta$$  is the area of the triangle  $$A B C$$ ) .

$$\bar { a } ,\bar { b } $$ and $$\bar { c } $$ are unit vector such that $$\bar { a } +\bar { b } -\bar { c } =0$$. then the angle between $$\bar { a } $$ and $$\bar { b } $$ is :-

Let $$ABCD$$ is a triangular pyramid with base vectors $$\vec {AB}= 2\bar {i}+3\bar {j}-\bar {k}$$ and $$\vec {AC}=\bar {i}-2\bar {k}$$, If volume of the triangular pyramid is $$\sqrt{150}$$ unit then its height is

If the vectors $$\bar { AB } =3\hat { i } +4\hat { k } $$ and $$\bar { AC } =5\hat { i } -2\hat j+4\hat k$$ are the sides of a triangle ABC, then the length of the median through A is: