Vector Algebra ...

A unit vector $$d$$ is equally inclined at an angle $$\alpha$$ with the vectors $$a=\cos \theta. i+ \sin \theta. j , b=-\sin \theta.i+\cos =\theta. j$$ and $$c=k$$. Then $$\alpha$$ is equal to 

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

The foot of the perpendicular drawn from a point with position vector $$\hat { i } +4\hat { k } $$ on the line joining the points $$\hat { j } +3\hat { k } $$, $$2\hat { i } -3\hat { j } +\hat { k } $$ is

The distance of the point $$  \text{P}  $$ with position vector  $$3\hat{i}+6 \hat{j}+8\hat{k}  $$ from $$  y  $$ - axis 

If  $$\vec { a } , \vec { b } , \vec { c }$$  are unit vectors such that  $$\vec { a } + \vec { b } + \vec { c } = 0 ,$$  the value of  $$\vec { a } \cdot \vec { b } + \vec { b } \cdot \vec { c } + \vec { c } \cdot \vec { a }$$  is

If $$\left( \bar { a } -\bar { b }  \right) =\bar { \left( a \right)  } =\bar { \left( b \right)  } $$ where $$\bar { a } $$ and $$\bar { b } $$ are non zero vectors then the angle between $$\bar { a } -\bar { b } $$

Line passing through $$ (3,4,5)  $$ and $$ (4,5,6)  $$ has direction ratios $$  \ldots $$

If radius vector of a point varies with time $$t$$ as $$\vec { r } =\vec { b } t\left( 1-\alpha t \right) $$ where $$\vec { b } $$ is a constant vector and a is a positive constant, then its

let $$\bar { a } ,\bar { b } ,\bar { c } $$ are three mutually perpendicular unit vectors and a unit vector $$ \bar { r } $$ satisfying the equation $$\left( \bar { b } -\bar { c }  \right) \times \left( \bar { r } \times \bar { a }  \right) +\left( \bar { c } -\bar { a }  \right) \times \left( \bar { r } \times \bar { b }  \right) +\left( \bar { a } -\bar { b }  \right) \times \left( \bar { r } \times \bar { c }  \right) =0$$ then $$\bar { r } $$ is __________________.

Let $$\overline { a } =4\hat { i } +3\hat { j } -\hat { k } $$ and$$\overline { b } =2\hat { i } -6\hat { j } -3\hat { k } .$$ Then a unit vector $$\bot $$ to both $$\overline { a }$$ and $$\overline { b } $$is.