Vector Algebra ...

The position vector of the point which divides the join of points $$2 \vec a -3\vec b$$ and $$\vec a+\vec b$$ in the ratio $$3:1$$ is 

The value of $$\hat { i }. (\hat { j } \times \hat { k }) + \hat { j }. (\hat { 
i } \times \hat { k })+\hat { k }. (\hat { i } \times \hat { j })$$ is

The value of $$ \hat {i} .( \hat {j} \times \hat {k}) + \hat {j} . ( \hat {i} \times  \hat {k})  + \hat {k} .( \hat {i} \times  \hat {j}) $$

If $$ \overrightarrow {a} $$ is non zero vector of magnitude 'a ' and $$ \lambda $$ a nonzero scalar then $$ \lambda \overrightarrow {a} $$ is unit vector

Three vectors of magnitudes $$a,\ 2a,3a$$ meeting a point and three directions are along the diagonals of three adjacent faces of a cube. The magnitude of their resultant is

If $$\vec {x}$$ is a vector whose initial point divides the line joining $$5\hat{i}$$, and $$ 5\hat{j}$$ in the ratio $$\lambda :1$$ and  the terminal point is the origin. Also given $$\left | \vec {x} \right |\leq \sqrt{37}$$, then $$\lambda $$ belongs to

A scooterist follows a track on a ground that turns to his left by an angle 60$$^{0}$$ after every 400 m. Starting from the given point displacement of the scooterist at the third turn and eighth turn are :

$$\mathrm{l}\mathrm{n}$$ a triangle O$$\mathrm{A}\mathrm{B},\ \mathrm{E}$$ is the mid-point of $$\mathrm{O}\mathrm{B}$$ and $$\mathrm{D}$$ is a point on $$\mathrm{A}\mathrm{B}$$ such that $$\mathrm{A}\mathrm{D}$$: $$\mathrm{D}\mathrm{B}=2: 1$$. lf $$\mathrm{O}\mathrm{D}$$ and $$\mathrm{A}\mathrm{E}$$ interesect at $$\mathrm{P}$$, then the ratio $$\displaystyle\frac{OP}{PD}$$ is

In a quadrilateral $$PQRS,\ \vec{PQ}=\vec{a}, \vec{QR}=\vec{b}, \vec{SP}=\vec{a} - \vec{b}.\ M$$ is the mid-point of $$QR$$ and $$X$$ is a point on $$SM$$ such that $$\vec{SX}=\dfrac{4}{5}\vec{SM}$$, then $$\vec{PX}$$ is

The position vectors of $$A$$ and $$B$$ are $$2\hat{i}+2\hat{j}+\hat{k}$$ and $$2\hat{i}+4\hat{j}+4\hat{k}.$$ The length of the internal bisector of $$\angle BOA$$ of the triangle $$AOB$$ is