Vector Algebra ...

If $$A, B$$ are two points on the curve $$y=x^{2}$$ in the $$x-y$$ plane satisfying $$\vec{OA}.\hat{i}=1$$ and $$\vec{OB}.\hat{i}=-2$$ then the length of the vectors $$2\vec{OA}-3\vec{OB}$$ is

In a tetrahedron if two pairs of opposite edges are at a right angles then the third pair is inclined at an angle of

If $$\overrightarrow { a } .\overrightarrow { b } =0$$ and also $$\overrightarrow { a } \times \overrightarrow { b } =0,$$ then

Let $$G$$ be the centroid of $$\triangle ABC$$. If $$\overrightarrow{AB}=\overrightarrow{a}$$ and $$\overrightarrow{AC}=\overrightarrow{b},$$ then $$\overrightarrow{AG}$$, in terms of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ is

If $$\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } $$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then the centroid of the triangle satisfies which of the following relation?

If $$\alpha(\vec a \times \vec b)+\beta(\vec b \times \vec c)+\gamma(\vec c \times \vec {a})=\vec{0}$$ and at least one of the scalars $$\alpha,\ \beta,\gamma$$ is non-zero, then the vectors $$\vec{a},\vec{b},\vec{c}$$ are

If $$I$$ is the center of a circle inscribed in a triangle $$ABC$$, then $$|BC|IA+|CA|IB+|AB|IC$$

lf the four points $$\overline{a},\overline{b},\overline{c},\overline{d}$$ are coplanar then $$[\overline{b}\overline{c}\overline{d}]+[\overline{c}\overline{a}\overline{d}]+[\overline{a}\overline{b}\overline{d}]=$$

A point $$O$$ is the centre of a circle circumscribed about a triangle $$ABC$$, then $$\vec{OA}\sin 2A + \vec{OB}\sin 2B + \vec{OC} \sin 2C $$ is equal to

If $$\vec {a}=x\hat {i}+12\hat {j}-\hat {k},\vec {b}=2\hat {i}+2x\hat {j}+\hat {k}$$ and $$\vec {c}=\hat {i}+\hat {k}$$ and given that the vectors $$\vec {a},\vec {b},\vec {c}$$ form a right handed system, then the range of $$x$$ is