Vector Algebra ...

If the vectors $$4\hat{i}-7\hat{j}-2\hat{k},\: \hat{i}+5\hat{j}-3\hat{k},\: 3\hat{i}-\lambda\hat{j}+\hat{k}$$ form a triangle then $$\lambda=$$

lf $$\vec{a}$$ and $$\vec{b}$$ are two non-parallel unit vectors and the vector $$\alpha\vec{a}+\vec{b}$$ bisects the internal angle between $$\vec{a}$$ and $$\vec{b}$$, then $$\alpha$$ is

lf $$A=(-3,2,5), B=(-3,4,5)$$ and $$C=(-3,4,7)$$ are the position vectors of vertices of $$\Delta ABC$$ then its circumcentre is

$$\hat { AB } =-3{ \hat { i }  }+4{ \hat { k }  }$$ and $$\hat { BC } =-\overline { i } -2\overline { k } $$ are the sides of the triangle $$ ABC$$ then the length of the median $$AM$$ is

Two forces act at the vertex $$\mathrm{A}$$ of quadrilateral $$\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$$ represented by $$\overline{AB},\ \overline{AD}$$ and two at $$\mathrm{C}$$ represented by $$\overline{CD}$$ and $$\overline{CB}$$. If $$\mathrm{E},\ \mathrm{F}$$ are mid points of $$\overline{AC}$$ and $$\overline{BD}$$ respectively, then their resultant is

If point $$O$$ is the centre of a circle circumscribed about a triangle $$ABC$$. Then $$\overline{OA}\sin 2A+\overline{OB}\sin2B+\overline{OC}\sin 2C=$$

Let $$G$$ and $$G^{1}$$ be the centroids of the triangles $$ABC$$ and $$A^{1}B^{1}C^{1}$$ respectively, then $$AA^{1}+BB^{1}+CC^{1}$$ is equal to

The ratio in which $$\overline{i}+2\overline{j}+3\overline{k}$$ divides the join $$\mathrm{o}\mathrm{f}-2\overline{i}+3\overline{j}+5\overline{k}$$ and $$7\overline{i}-\overline{k}$$ is

Orthocentre of an equilateral triangle $$ABC$$ is the origin $$\mathrm{O}$$. If $$A=\overline{a},\ B=\overline{b},\ C=\overline{c}$$ then $$\overline{AB}+2\overline{BC}+3\overline{CA}=$$