Vector Algebra ...

If $$M$$ and $$N$$ are the mid-points of the diagonals $$AC$$ and $$BD$$ respectively of a quadrilateral $$ABCD$$, then the value of $$\overline { AB } +\overline { AD } +\overline { CB } +\overline { CD } $$

If $$\overline { a } $$ and $$\overline { b } $$ include an angle of $${120}^{o}$$ and their magnitudes are $$2$$ and $$\sqrt{3}$$ then $$\overline { a } .\overline { b } $$ is

The perimeter of the triangle whose vertices are the points $$2\bar{i}-\bar{j}+\bar{k}, \bar{i}-3\bar{j}-5\bar{K},3\bar{i}-4\bar{j}-4\bar{j}-4\bar{k}$$ is 

If a,b,c are all non-zero, then the number of values of $$\lambda $$ such that $$a(-4\bar{i}+5\bar{j})+b(3\bar{i}-3\bar{j}+\bar{k})+c(\bar{i}+\bar{j}+3\bar{k})=\lambda (a\bar{i}+b\bar{j}+c\bar{k})$$ is :

If $$S$$ is the circumcentre, $$O$$ is the orthocentre of $$\triangle{ABC}$$, then $$\overline { SA } +\overline { SB } +\overline { SB } $$ equals

If $$\overline { a } =\left( 2\overline { i } -10\overline { j } +6\overline { k }  \right) ;\overline { b } =\left( 5\overline { i } -3\overline { j } +\overline { k }  \right) $$. The ratio of projection of $$\overline { a } $$ on $$\overline { b } $$ to projection of $$\overline { b } $$ on $$\overline { a } $$ is

Let $$\bar{a},\bar{b}$$ be two noncollinear vectors. If $$A=(x+4y)\bar{a}+(2x+y+1)\bar{b},$$
$$B=(y-2x+2)\bar{a}+(2x-3y-1)\bar{b} \quad and \quad 3A=2B$$ then (x,y) =

If $$\overline { a } $$ is a vector of magnitude $$\sqrt{3}$$ and $$\overline { b } $$ is unit vector making an angle $$\tan ^{ -1 }{ \left( 1/\sqrt { 2 }  \right)  } $$ with $$\overline { a } $$ then projection of $$\overline { a } $$ on $$\overline { b } $$ is

Given $$\vec{\alpha} = 3\hat{i} + \hat{j} + 2\hat{k}\ ,\ \vec{\beta} = \hat{i} - 2\hat{j} - 4\hat{k}$$ are the position vectors of the points $$A$$ and $$B$$. Then the distance of the point $$-\hat{i} + \hat{j} + \hat{k}$$ from the passing through $$B$$ and perpendicular to $$AB$$ is 

A (1,-1,-1) , B (2,1,-2) and C (-5,2,-6) are the position vectors of the vertices of triangle ABC  The length of the bisector of its internal angle at A is: