Vector Algebra ...

Given unit vectors  $$\hat {m}, \hat {n}$$ and $$\hat {p}$$ such that $$\left( \widehat { \hat { m } \hat { n }  }  \right) =\hat { p } \widehat {  } \left( \hat { m } \times \hat { n }  \right) =\alpha$$ then the value of $$[\hat { n } \hat { p } \hat { m } ]$$ in terms of $$\alpha$$ is :

The position vector of two points A and B are 6a+2b and a-3b. If a point C divides AB in the ratio 3 : 2, then the position vector of C is

$$\left| {\overline x } \right| = \left| {\overline y } \right| = 1,\,\overline x  \bot \overline y ,\,\left| {\overline x  + \overline y } \right| = $$

The position vector of a point $$C$$ with respect to $$B$$ is $$\hat { i } + \hat { j }$$ and that of B with respect to A is $$\hat { i } - \hat { j }$$. The position vector of $$C$$ with respect to $$A$$ is

D,E and F are the mid-points of the sides BC,CA and AB respectively of $$\Delta ABC$$ and G is the centroid of the triangle, then $$\vec{GD}+\vec{GE}+\vec{GF}=$$

If  $$\vec { u } =\vec { a } -\vec { b } ~;\vec{ v } =\vec { a } +\vec { b } ~~\&~ |\vec { a }| =|\vec { b }| =2,$$ then $$\left| \vec { u } \times \vec { \upsilon  }  \right| $$ is equal to:

Vector equation of the plane $$\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}+2\hat{j}+3\hat{k})$$ in the scalar dot product from is

$$\vec{r}.\hat{i}=2\vec{r}.\hat{j}=4\vec{r}.\hat{k}$$ and $$\left|\vec{r}\right|=\sqrt{84}$$, then $$\left|\vec{r}.\left(2\hat{i}-3\hat{j}+\hat{k}\right)\right|$$ is equal to 

If $$a,b$$ and $$c$$ are position vector of $$A,B$$ and $$C$$ respectively of $$\triangle ABC$$ and if $$|a-b|=4,|b-c|=2, |c-a|=3$$, then the distance between the centroid and incentre of $$\triangle ABC$$ is 

If the position vectors of $$A, B, C, D$$ are $$3\hat{i} + 2\hat{j} + \hat{k}, 4\hat{i} + 5\hat{j} + 5\hat{k}, 4\hat{i} + 2\hat{j} - 2\hat{k}, 6\hat{i} + 5\hat{j} - \hat{k}$$ respectively then the position vector of the point of intersection of $$\bar{AB}$$ and $$\bar{CD}$$ is