Vector Algebra ...

$$\vec{a} = 2 \widehat{i} - \widehat{j} + \widehat{k}, \vec{b} = \widehat{i} + 2\widehat{j} - \widehat{k} $$ and $$ \vec{c} = \widehat{i} + \widehat{j} - 2 \widehat{k}$$. A vector coplanar with $$\vec{b}$$ and $$\vec{c}$$ whose projection on $$\vec{a}$$ is magnitude $$\displaystyle \sqrt{\dfrac{2}{3}} $$ is

'$$P$$' is a point inside the triangle $$ABC$$, such that $$\displaystyle BC\left ( \vec{PA} \right )+CA\left ( \vec{PB} \right )+AB\left ( \vec{PC} \right )=0,$$ then for the triangle $$ABC$$ the point $$P$$ is its :

Let the pairs $$\vec{a}, \vec{b}$$ and $$\vec{c}, \vec{d}$$ each determine a plane, then the planes are parallel if

Let $$\vec{a},\vec{b},\vec{c}$$ be vectors of length $$3,4,5$$ respectively. Let $$\vec{a}$$ be perpendicular to $$\vec{b}+\vec{c},\vec{b}\,to\,\vec{c}+\vec{a}$$ and $$\vec{c}\,to\,\vec{a}+\vec{b}$$. Then $$\begin{vmatrix}\vec{a}+\vec{b}+\vec{c}\end{vmatrix}$$ is

$$ABCDEF$$ is a regular hexagon . The centre of hexagon is a point O. Then the value of 
$$\overrightarrow{AB}+ \overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}$$ is 

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors a, b, c such that a.b = b.c = c.a = 1/2. What is the volume of the parallelopipe.

Let $$b = 4i + 3j $$ and $$c$$ be two vectors perpendicular to each other in the xy-plane. If $$ r_i, \ i =1, 2 ... n$$, are the vectors in the same plane having projections $$1 $$ and $$2$$ along $$b$$ and $$c$$ respectively then $$ \displaystyle \sum_{i=1}^{n} \left| r_{i}\right|^{2}$$ is equal to

The magnitude of the projection of the vector $$\overline{a} =4\overline{i}-3\overline{j}+2\overline{k}$$ on the line which makes equal angles with the coordinate axes is 

Let $$ x_{0}$$ and $$x_{1}$$ be the critical points of $$\displaystyle f(x) =\int_{1}^{x}(t(t + 1) (t + 2) (t + 3)- 24).dt $$ and $$ \vec r$$ & $$\vec r'$$ be the parallel vectors with $$\left| \vec r \right| =\left| x_{0}\right| $$ and $$\left| \vec r\ ' \right| =\left|x_{1}\right| ,$$ then $$ \vec r\cdot \vec r\ '$$ is equal to

If one point on the vector $$2i -4j-k$$ is $$(2,1,3)$$,the other point is?