Vector Algebra ...

$$ABC$$ is a triangle, the point $$P$$ is on side $$BC$$ such that $$3\vec {BP}=2\vec {PC}$$, the point $$Q$$ is on the line $$\vec {CA}$$ such that $$4\vec {CQ}=\vec {QA}$$. If $$R$$ is the common point of $$\vec {AP}$$ & $$\vec {BQ}$$, then the ratio in which the line joining $$CR$$ divides $$\vec {AB}$$ is

If $$\hat {i},\hat {j},\hat {k}$$ are positive vectors of $$A,B,C$$ and $$\vec {AB}=\vec {CX}$$, then positive vector of $$X$$ is

Four persons $$P,\ Q,\ R$$ and $$S$$ are initially at the four corners of a square side $$d$$. Each person now moves with a constant speed $$v$$ in such a way that $$P$$ always moves directly towards $$Q,\ Q$$ towards $$R,\ R$$ towards $$S,$$ and $$S$$ towards $$P$$. The four persons will meet after time 

If $$\bar a + \bar b$$ is perpendicular to $$\bar b$$ and $$\bar a + 2\bar b$$ is perpendicular to $$\bar a$$ then. 

The position vector of point P, is?

If $$\overline { a } =\cfrac { 1 }{ \sqrt { 10 }  } \left( 3\overline { i } +\overline { k }  \right) ;\overline { a } =\cfrac { 1 }{ 7 } \left( 2\overline { i } +3\overline { j } -6\overline { k }  \right) $$ then the value of
$$(2\overline { a } -\overline { b } ).[(\overline { a } \times \overline { b } )\times (\overline { a } +2\overline { b } )]$$

If $$\vec a,\vec b,\vec c$$ non-zero vectors such that $$\vec a$$ is perpendicular to $$\vec b$$ and $$\vec c$$ and non-zero  vector coplanar with $$\vec a + \vec b$$ and $$2\vec b - \vec c$$ and $$\vec d.\vec a = 1$$ , then the minimum value of $$\left| {\vec d} \right|$$

Let $$\vec{a},\vec{b},\vec{c}$$ be three non-zero vectors such that $$\vec{a}+\vec{b}+\vec{c}=0$$ and $$\lambda \vec{b}\times \vec{a}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}=0$$, then $$\lambda$$ is

If $$a,b,c \in N$$, the number of points having position vectors $$a\hat i + b\hat j + c\hat k$$ such that $$6 \le a + b + c \le 10$$ is

If $$\vec{a}$$ be any vector, then the value of $$|\vec{a} \times \hat{ i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$$