Vector Algebra ...

Let $$\vec{A}=2\hat{i}+7\hat{j},\vec{B}=\hat{i}+2\hat{j}+4\hat{k},\ \displaystyle \vec{C}=\dfrac{9\hat{i}+30\hat{j}+4\hat{k}}{5}$$. 

If the position vectors of $$A,B,C$$ are $$3\hat{i}+\hat{j}-\hat{k}$$, $$\hat{i}-2\hat{j}$$, $$2\hat{i}-\hat{j}{+}3\hat{k}$$ then the position vector of the centroid of the triangle $$ABC$$ is

If $$\vec a, \vec b, \vec c, \vec d$$ are the position vectors of the points $$A, B, C, D$$ respectively such that $$3\vec{a}+5\vec{b}-3\vec{c}-5\vec{d}=0$$ then $$AB$$ intersects $$CD$$ in the ratio

If $$(2, -1, 2)$$ is the centroid of tetrahedron $$OABC$$ and $$G_{1}$$ is the centroid of $$\Delta ABC$$ then $$|\overline{OG}_{1}|=$$

If $$\overline{a}=\overline{i}+\overline{j}+\overline{k},\overline{b}=2\overline{i}-3\overline{j}+\overline{k}$$, then $$\displaystyle \dfrac{\overline{a}\times\overline{b}}{|\overline{a}\times\overline{b}|}+\dfrac{\overline{b}\times\overline{a}}{|\overline{b}\times\overline{a}|}=$$

The position vectors of $$A, B , C, D$$ are $$\overline{a},\overline{b},\ \overline{c},\overline{d}$$ respectively and $$|\overline{a}-\overline{d}|=|\overline{b}-\overline{d}|=|\overline{c}-\overline{d}|$$, then for the triangle $$ABC$$, $$D$$ is

Let $$\overrightarrow { b } =4\hat { i } +3\hat { j } $$ and $$\overrightarrow{c}$$ be two vector perpendicular to each other in the $$xy$$-plane. Then a vector in the same plane having projections $$1$$ and $$2$$ along $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$, respectively, is

The angle between the Vectors $$\vec{a}\times\vec{b}$$ and $$\vec{b}\times\vec{a}$$ is

The vectors $$\vec{a},\ \vec{b},\ \vec{a}\times \vec{b}$$ form