Vector Algebra ...

If the vertices of a $$\Delta ABC$$ are $$A= (1,-1,-3)$$ , $$B= (2, 1, -2)$$ and $$C=(-5,2,-6)$$ then the length of the internal bisector of angle $$A$$ is

The condition for the vectors $$\vec{a},\vec{b},\vec{c},\vec{d}$$ to be the sides of a parallelogram taken in order is

Let $$A$$ and $$B$$ be points with position vectors $$\overline{a}$$ and $$\overline{b}$$ with respect to origin $$O$$. If the point $$C$$ on $$OA$$ is such that $$2\overline{AC}=\overline{CO}$$, $$\overline{CD}$$ is parallel to $$\overline{OB}$$ and $$|\overline{CD}|=3|\overline{OB}|$$ then $$AD$$ is

The adjacent sides of a parallelogram are $$2{ \hat { i }  }+4{ \hat { j }  }-5{ \hat { k }  }$$ and $$ { \hat { i }  }+2{ \hat { j }  }{ + }3{ \hat { k }  }$$ then the unit vector parallel to a diagonal is

If the diagonals of a parallelogram are $$\overline{i}+5\overline{j}-2\overline{k}$$ and $$-2\overline{i}+\overline{j}+3\overline{k}$$, then the lengths of its sides are

Let $$A({\vec{a}})$$ , $$B({\vec{b}}), C({\vec{c}})$$ be the vertices of the triangle $$ABC$$ and let $$DEF$$ be the mid points of the sides $$BC, CA, AB$$ respectively. If $$P$$ divides the median $$AD$$ in the ratio $$2:1$$ then the position vector of $$P$$ is

If $$I$$ is the centre of a circle inscribed in a triangle $$ABC$$, then $$|\overline{BC}|\overline{IA}+|\overline{CA}|\overline{IB}+|\overline{AB}|\overline{IC}$$ is

Let $$A=2\hat{i}+4\hat{j}-\hat{k}$$, $$B=4\hat{i}+5\hat{j}{+}\hat{k}$$. If the centroid $$G$$ of the triangle $$ABC$$ is $$3\hat{i}+5\hat{j}-\hat{k}$$, then the position vector of $$C$$ is

If $$\vec{{r}} {\times} \vec{{a}}=\vec{{b}}{\times}\vec{{a}};\ \vec{{r}}{\times}\vec{{b}}=\vec{{a}}{\times}\vec{{b}};\ \vec{a}\neq 0,\vec{b}\neq 0,\vec{a}\neq\lambda\vec{b};\ \vec{a}$$ is not perpendicular to $$\vec{b}$$, then $$\vec{r}=$$

The plane $$2x-3y+z+6=0$$ divides the line segment joining $$(2, 4, 16)$$ and $$(3, 5, -4)$$ in the ratio