Vector Algebra ...

The vector sum of $$N$$ coplanar forces, having magnitude of $$F$$, when each force is making an angle of $$\displaystyle \frac{2\pi}{N}$$ with that preceding it, is:

Five equal forces each of 20N are acting at a point in the same plane. If the angles between them are same, then the resultant of these forces is:

If there are 11 vectors each having a magnitude equal to $$| \vec{p} |$$ and if each side of polygon subtends an angle $$30^{0}$$ at the centre of the polygon. Then the resultant is 

$$\mathrm{If}$$ $$\vec{AD},\ \vec{BE},\ \vec{CF}$$ are medians of an equilateral triangle $$\mathrm{A}\mathrm{B}\mathrm{C}$$, then $$\vec{AD}+\vec{BE}+\vec{CF}$$ equals to 

If $$\overrightarrow { a } ,\overrightarrow { b } $$ and $$\overrightarrow { c } $$ are three non-zero vectors such that $$\overrightarrow { a } .\overrightarrow { b } =\overrightarrow { a } .\overrightarrow { c } ,$$ then

Let $$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$ be a trapezium and let $$\mathrm{P},\mathrm{Q}$$ be the midpoints of the nonparallel sides $$\mathrm{A}\mathrm{D},\ \mathrm{B}\mathrm{C}$$ respectively. Then $$\vec { PQ } $$

Let $$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$ be a parallelogram and let $$\mathrm{L}$$ and $$\mathrm{M}$$ be the midpoints of the sides $${ BC } $$ and $$ { CD }$$  respectively. Then  $$\vec { AL } +\vec { AM } =$$

$${A}{B}{C}{D}$$ is a quadrilateral, $$E$$ is the point of intersection of the line joining the middle points of the opposite sides. lf $$O$$ is any point, then $$\hat { OA } +\hat { OB } +\hat { OC } +\hat { OD } =$$

If $$\overline{DA}=\overline{a};\overline{AB}=\overline{b}$$ and $$\overline{CB}=k\overline{a}$$ where $$k>0$$ and $$x, y$$ are the midpoints of $$DB$$ and $$AC$$ respectively such that $$|\overline{a}|=17$$ and $$|\overline{XY}|=4$$, then $$\mathrm{k}=$$

The incentre of the triangle formed by the points $$ { \hat { i }  }+{ \hat { j }  }+{ \hat { k }  },$$ $$ 4{ \hat { i }  }+{ \hat { j }  }+{ \hat { k }  }$$, and $$4{ \hat { i }  }+5{ \hat { j }  }+\hat { k }$$  is