Vector Algebra ...

$$ABCD$$ is a quadrilateral, $$E$$ is the point of intersection of the line joining the midpoints of the opposite sides. If $$O$$ is any point and $$\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = \vec{x OE},$$ then $$x$$ is equal to

If $$\overrightarrow{b}$$ is a vector whose initial point divides the join of $$5\widehat{i}$$ and $$5\widehat{j}$$ in the ratio $$k : 1$$  and whose terminal point is the origin and $$|\vec b| \leq \sqrt{37}$$, then $$k$$ lies in the interval

Vectors $$\vec a = \hat i + 2 \hat j + 3 \hat k, \vec b = 2 \hat i - \hat j + \hat k$$ and $$\vec c = 3 \hat i + \hat j + 4 \hat k$$ are so placed that the end point of one vector is the starting point of the next vector, then the vectors are

$$L_{1}and L_{2}$$ are two lines whose vector equations are

$$L_{1}:\vec{r}=\lambda \left ( (\cos \theta+\sqrt{3})\hat{i}+(\sqrt{2}\sin\theta)\hat{j}+(\cos \theta-\sqrt{3})\hat{k} \right )$$

$$L_{2}:\vec{r}=\mu \left ( a \hat{i}+b \hat{j}+c\hat{k} \right ),$$

Where $$\lambda\ and\ \mu $$ are scalars and $$\alpha$$ is the acute angle

between $$L_{1}\ and\ L_{2}$$ . If the angle '$$\alpha$$'is independent

of $$\theta $$ then the value of '$$\alpha$$ ' is

$$\overrightarrow{AR}$$ is

$$ABCD$$ a parallelogram, $$A_1$$ and $$B_1$$ are the midpoints of sides $$BC$$ and $$CD$$, respectively. If $$\vec{AA_1} + \vec{AB_1} = \lambda \vec{AC}$$, then $$\lambda$$ is equal to

In a parallelogram $$OABC,$$ vectors $$\vec{a}, \vec{b}, \vec{c}$$ are, respectively, the position vectors of vertices $$A, B, C$$ with reference to $$O$$ as origin. A point $$E$$ is taken on the side $$BC$$  which divides it in the ratio of $$2 : 1$$. Also, the line segment  $$AE$$  intersects the line bisecting the angle $$\angle$$AOC internally at point $$P$$. If $$CP$$ when extended meets $$AB$$  in point $$F,$$  then the position vector of point $$P$$  is

The ratio $$\displaystyle \dfrac{OX}{XC}$$ is

The projection of the line joining the points $$(3, 4, 5)$$ and $$(4, 6, 3)$$ on the line joining the points $$(-1, 2, 4)$$ and $$(1, 0, 5)$$ is

A parallelogram is constructed on the vectors $$\bar{\alpha }$$ and $$\bar{\beta }$$. A vector which coincides with the altitude of the parallelogram and perpendicular to the side $$\bar{\alpha }$$ expressed in terms of the vectors $$\bar{\alpha }$$ and $$\bar{\beta }$$ is