Vector Algebra ...

Let $$\overrightarrow{a},\overrightarrow{b}$$ be the position vectors of points $$A$$ and $$B$$ with respect to $$O$$ and $$\left|\overrightarrow{a}\right|=a,\left|\overrightarrow{b}\right|=b$$ the points $$C$$ and $$D$$ divides $$A$$ internally and externally in the ratio $$2:3$$ If $$\overrightarrow{OC}$$ and $$\overrightarrow{OD}$$ are perpendicular then

If $$\vec{a}+\vec{b}+\vec{c}=vec{0}$$ then $$\vec{a}\times \vec{b}=?$$

The ratio in which $$i+2j+3k$$ divides the join of $$-2i+3j+5k$$ and $$7i-k$$ is?

Let $$\left| \overline { a } +\overline { b }  \right| =\left| \overline { a } -\overline { b }  \right| $$. If $$\left| \overline { a } \times \overline { b }  \right| =\lambda \left| \overline { a }  \right| $$, then $$\lambda=$$

Let $$A,B,C$$ be distinct point with position vectors $$\hat{i}+\hat{j}$$, $$\hat{i}-\hat{j}$$, $$p\hat{i}-q\hat{j}+r\hat{k}$$ respectively. Points $$A,B,C$$ are collinear, then which of the following can be correct:

The projection of the vector $$\hat{i}-2\hat{j}+\hat{k}$$ on he vector $$4\hat{i}-4\hat{j}+7\hat{k}$$ is equal to:

If $$\left|\overrightarrow{a} \right|=1$$, the projection of $$\overrightarrow{r}$$ along $$\overrightarrow{a} $$ is $$2$$ and $$\overrightarrow{a}\times \overrightarrow{r}+\overrightarrow{b}=\overrightarrow{r}$$, then $$\overrightarrow{r}=$$

If $$\overrightarrow{u}, \overrightarrow{v}$$ and $$\overrightarrow{w}$$ are three non-coplanar vectors, then $$(\overrightarrow{u} + \overrightarrow{v} - \overrightarrow{w}). (\overrightarrow{u} - \overrightarrow{v}) \times (\overrightarrow{v} - \overrightarrow{w})$$ equals

If $$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}\neq 0$$ then $$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=$$

A unit vector perpendicular to the plane of the triangle ABC with the position vectors  $$\vec a\,\,\,\,\vec b\,\,\vec c$$ of the vectors A,B,C, is