Vector Algebra ...

In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC, AE=3EC,$$ Let $$P$$ be the point of intersection of $$AD$$ and $$BE$$. Then $$\dfrac{BE}{PE}=$$

If $$\overline{a}+\overline{b}+\overline{c}=\alpha\overline{d},\overline{b}+\overline{c}+\overline{d}=\beta\overline{a}$$, then $$\overline{a}+\overline{b}+\overline{c}+\overline{d}$$ is 

If $$OABC$$ is a parallelogram with $$\vec{OB}=\vec{a},\vec{AB}=\vec{b}$$ then $$\vec{OA}=$$

Let us define, the length of a vector $$a\overline{i}+b\overline{j}+c\overline{k}$$ as $$|{a}|+|{b}|+|{c}|$$. This definition coincides with the usual definition of the length of a vector $$a\overline{i}+b\overline{j}+c\overline{k}$$ if

In $$\Delta ABC,\ D,\ E,\ F$$ are midpoints of the sides $$BC, CA$$ and $$AB$$ respectively. $$O$$' is the circumcentre, $$G$$' is the centroid, $$H$$' is the orthocentre and $$P$$ is any point.
Match the following

Vector area is a vector quantity associated with each plane figure whose magnitude is

Let $$OABC$$ be a parallelogram and $$D$$ the midpoint of $$OA$$. The ratio in which $$OB$$ divides $$CD$$ in the ratio

In $$\Delta OAB$$, if $$\vec{OA}=\vec{a},\ \vec{OB}=\vec{b}.  L$$ is mid point of $$\vec{OA}$$ and $$M$$ is point on $$\vec{OB}$$ such that $$\vec{OM}:\vec{MB}=2:1$$. If $$P$$ is mid point of $$LM$$ then $$\vec{AP}=$$

If the vectors $$\vec{c},\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$$ and $$\vec{b}=\hat{j}$$ are such that $$\vec{a},\vec{c}$$ and $$\vec{b}$$ form a right handed system, then $$\vec{c}=$$

The vector $$\vec{AB}=3\hat{i}+4\hat{k}$$ and $$\vec{AC}=5\hat{i}-2\hat{j}+4\hat{k}$$ are the sides of a $$\Delta ABC$$ where $$A$$ is the origin. The length of median through $${A}$$ is