Vector Algebra ...

If $$\overrightarrow{a} = 2\hat{i} - \hat{j} + \hat{k}$$, $$\overrightarrow{b} = \hat{i} + 2\hat{j} - \hat{k}$$, $$\overrightarrow{c} = \hat{i} + \hat{j} - 2\hat{k}$$, then a vector in the plane of $$\hat{b}$$ and $$\hat{c}$$ whose projection on $$\hat{a}$$ is a magnitude of $$\sqrt{\frac{2}{3}}$$ is 

Let $$\vec { p } $$ and $$\vec { q } $$ be the position vectors of the points $$P$$ and $$Q$$ respectively with respect to origin $$O$$. The points $$R$$ and $$S$$ divide $$PQ$$ internally and externally respectively in the ratio $$2:3$$. If $$\overrightarrow { OR } $$ and $$\overrightarrow { OS } $$ are perpendicular, then which one of the following is correct?

A unit vector a makes an angel $$\Pi /4$$ with the z-axis. If a+i+j is a unit vector, then a can be equal to  

Let $$\vec{u},\ \vec{v},\ \vec{w}$$ be such that $$\left| \vec { u }  \right| =1$$,  $$\left| \vec { v }  \right| =2$$, $$\left| \vec { w }  \right| =3$$. If the projection of $$\vec{v}$$ along $$\vec{u}$$ is equal to projection of $$\vec{w}$$ along $$\vec{u}$$ and $$\vec{v}$$ and $$\vec{w}$$ are perpendicular to each other then $$\left| \bar { u } -\bar { v } +\bar { w }  \right|$$ equals-

If $$ABCDE$$ is a pentagon, then $$\vec {AB}+\vec {AE}+\vec {BC}+\vec {DC}+\vec {ED}+\vec {AC}$$ equals

Let $$\overline {a}, \overline {b}$$ be two noncollinear vectors. If $$\overline {OA}=(x+4y)\overline {a}+(2x+y+1)\overline {b}, \overline {OB}=(y-2x+2)\overline {a}+(2x-3y-1)\overline {b}$$ and $$3\overline {OA}=2\overline{OB}$$, then $$(x,y)=$$

The value of $$|\overrightarrow A  + \overrightarrow B  - \overrightarrow C  + \overrightarrow D |$$ can be zero if :-

The projection of $$\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$$ on the vector $$\vec{b}=\hat{i}+2\hat{j}-\hat{k}$$ is?

The set of values of $$'c'$$ for which the angle between the vectors $$(cx\hat{i}-6\hat{j}+3\hat{k})$$  and  $$(x\hat{i}-2\hat{j}+2cx\hat{k})$$ is acute for every $$x\in R$$  is

A vector $$\vec{a}$$ has components $$2$$ p and $$1$$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $$\vec{a}$$ components $$p+1$$ and $$1$$, then?