Vector Algebra ...

$$AB=3i+j-k$$ and $$AC=i-j+3k$$. If the point $$P$$ on the line segment $$BC$$ is equidistant from $$AB$$ and $$AC,$$ then $$AP$$ is

If $$4i+7j+8k, 2i+7j+7k$$ and $$3i+5j+7k$$ are the position vectors of the vertices $$A,B$$ and $$C$$ respectively of triangle $$ABC$$. The position vector of the point where the bisector of angle $$A$$ meets $$BC.$$

A straight line $$'L'$$ cuts the sides $$AB,AC$$ and $$AD$$ of a parallelogram $$ABCD$$ at points $${ B }_{ 1 },{ C }_{ 1 }$$ and $${D}_{1}$$ respectively. If $$\overrightarrow { A{ B }_{ 1 } } ={ \lambda  }_{ 1 }\overrightarrow { AB } ,\overrightarrow { A{ D }_{ 1 } } ={ \lambda  }_{ 2 }\overrightarrow { AD } $$ and $$\overrightarrow { A{ C }_{ 1 } } ={ \lambda  }_{ 3 }\overrightarrow { AC } $$, then $$\displaystyle \frac { 1 }{ { \lambda  }_{ 3 } } $$ is equal to

The sum of the three vectors determined by the medians of a triangle directed from the vertices is

In a parallelogram $$ABCD$$, $$\left| AB \right| =a,\left| AD \right| =b$$ and $$\left| AC \right| =c$$. Then $$DB.AB$$ has the value

If $$O$$ and $$O'$$ are circumcenter and orthocenter of a triangle $$ABC$$ then $$\left( OA+OB+OC \right) $$ equals

The position vectors of three consecutive vertices of a parallelogram are $$i+j+k, i+3j+5k$$ and $$7i+9j+11k$$. The position vector of the fourth vertex is

Let $$G$$ be the centroid of a triangle $$ABC$$. If $$AB=a,AC=b$$ then the bisector $$AG$$, in terms of vectors $$a$$ and $$b$$ is

The value of $$\vec i \times (\vec a \times \vec i) + \vec j \times (\vec a \times \vec j) + \vec k \times (\vec a \times \vec k)$$ is (where $$\vec i, \vec j, \vec k$$ are unit vectors)

Let $$\vec{a}=\hat{i}+\hat{j}+3\hat{k}\;\&\;\vec{b}=2\hat{i}-3\hat{j}+4\hat{k}$$. If projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\displaystyle\frac{k}{\sqrt{29}}$$, then the value of $$(k-2)$$ is