Vector Algebra ...

The position vectors of two vertices and the centroid of a triangle are $$\vec{i}+\vec{j}$$, $$2\vec{i}-\vec{j}+\vec{k}$$ and $$\vec{k}$$ respectively, then the position vector of the third vertex of the triangle is 

$$\displaystyle R(\bar{r})$$ is any point on a semi-circle, $$\displaystyle P(\bar{p})$$ and $$\displaystyle Q(\bar{q})$$ are the position vector of the end point of the diameter of that semi-circle, then $$\displaystyle \overline{PR}\cdot \overline{QR}$$ is equal to

In a triangle OAB, E is the mid-point of OB and D is a point on AB such that AD: DB = 2 : 1. If OD and AE intersect at P, determine the ratio OP : PD using vector methods.

In a triangle ABC, D divides BC in the ratio 3 : 2 and E divides CA in the ratio 1 : 3. The lines AD and BE meet at H and CH meets AB in F. Find the ratio in which F divides AB.

The difference of the squares on the diagonals is four times the rectangle contained by either of these sides and the projection of the other upon it.

$$ABC$$ is a $$\displaystyle \Delta $$ and $$G$$ is its centroid. If $$\displaystyle \overline{AB}= \bar{b}$$ and $$\displaystyle \overline{AC}= \bar{c}$$, then $$\displaystyle \overline{AG}$$ is equal to

$$\displaystyle \bar{a}= x\hat{i}+y\hat{j}+z\hat{k},\bar{b}= \hat{j}$$ then the vector $$\displaystyle \bar{c}$$ for which $$\displaystyle \bar{a},\bar{b},\bar{c}$$ form a right hand triad

Let $$\displaystyle \bar{p}$$ and $$\displaystyle \bar{q}$$ be two distinct points. Let $$R$$ and $$S$$ be the points dividing $$PQ$$ internally and externally in the ratio $$2:3$$. If $$\displaystyle \overline{OR} \perp  \overline{OS},$$ then

Let $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ be vectors with magnitudes $$3, 4$$ and $$5$$ respectively and $$\bar{a}+\bar{b}+\bar{c}=0$$, then the value of $$\bar{a}.\bar{b}+\bar{b}.\bar{c}+\bar{c}.\bar{a}$$ is

The sides of a $$\triangle$$ are in A.P, then the line joining the centroid to the incenter is parallel to