Vector Algebra ...

If $$\bar{\alpha }$$, $$\bar{\beta }$$ and $$\bar{\gamma }$$ be vertices of a $$\triangle $$ whose circumcenter is at the origin, then orthocenter is given by

If $$S$$ is the circumcenter, $$O$$ is the orthocenter of $$\triangle ABC$$, then $$\displaystyle \vec{SA}+\vec{SB}+\vec{SC}= $$

For non-zero vectors $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$, $$\left | \left ( \bar{a}\times \bar{b} \right ).\bar{c} \right |=\left | \bar{a} \right |\left | \bar{b} \right |\left | \bar{c} \right |$$ iff

Two planes are perpendicular to each other,one of them contains vector $$\bar{a}$$ and $$\bar{b}$$, other contains $$\bar{c}$$ and $$\bar{d}$$ then $$\left ( \bar{a}\times \bar{b} \right )\cdot \left (\bar{c}\times \bar{d}  \right )=$$

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

If a. $$\displaystyle b\neq 0,$$ find the vector r which satisfies the equations $$\displaystyle \left ( r-c \right )\times b= 0, r.a= 0$$

If $$\alpha \bar{a}+\beta \bar{b}+\gamma \bar{c}=0$$, then $$\left ( \bar{a}\times \bar{b} \right )\times \left [ \left ( \bar{b}\times \bar{c} \right )\times \left ( \bar{c}\times \bar{a} \right ) \right ]$$ is equal to

The position vector of the points $$A$$ and $$B$$ are respectively $$\bar{a}$$ and $$\bar{b}$$  divides $$AB$$ in the ratio $$3:1$$ and $$Q$$ iis the midpoint of $$AP$$. The position vector of $$Q$$ is

For three unit vectors $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$, if $$\bar{a}+\bar{b}+\bar{c}= \bar{0}$$, then the value of $$\bar{a}\cdot \left ( \bar{b}+\bar{c} \right )+\bar{b}\cdot \left ( \bar{c}+\bar{a} \right )+\bar{c}\cdot \left ( \bar{a}+\bar{b} \right )$$ is equal to

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $$\bar{OA}+\bar{OB}+\bar{OC}+\bar{OD}=$$