If $\vec{PO} + \vec{OQ} = \vec{QO} + \vec{OR}$,then

If the vectors $-3 \hat{i} + 4 \hat{j} + \lambda \hat{k}$ and $\mu \hat{i} + 8 \hat{j} + 6 \hat{k}$ are collinear, then $\lambda - \mu =$

Let $\vec{v}$ be a vector in the plane such that $|\vec{v} - \hat{i}| = |\vec{v} - 2\hat{j}| = |\vec{v} - \hat{j}|$. Then,$|\vec{v}|$ lies in the interval

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$,$\vec{b}=2 \hat{i}+3 \hat{j}-5 \hat{k}$ and $\vec{c}=3 \hat{i}-\hat{j}+\lambda \hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b}+\vec{c}$. If $\vec{r} \cdot \vec{a}=3$,then $3 \lambda$ is equal to :

If the vector $19 \hat{i}+22 \hat{j}+5 \hat{k}$ bisects an angle between the vectors $a$ and $6 \hat{i}+8 \hat{j}$, then the unit vector in the direction of $a$ is

If $2 \hat{i}+\hat{j}-\hat{k}$,$\hat{i}-3 \hat{j}+5 \hat{k}$ and $-3 \hat{i}+4 \hat{j}+4 \hat{k}$ are the position vectors of three points $A$,$B$ and $C$ respectively,then