If $A(2, 3, 5)$,$B(1, 2, 3)$,$C(-5, 4, -2)$,and $D(1, 10, 10)$,then ...

Let $O$ be the origin and the position vectors of $A$ and $B$ be $2 \hat{i}+2 \hat{j}+\hat{k}$ and $2 \hat{i}+4 \hat{j}+4 \hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$,then the length of $OC$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}$ be two vectors such that $\vec{a} \cdot \vec{b}=1$, $\cos(\theta) = \frac{1}{3}$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, and the components of $\vec{b}$ with respect to $(\hat{i}, \hat{j}, \hat{k})$ are integers. Then the number of possible vectors that represent $\vec{b}$ is

If $\bar{a}, \bar{b}, \bar{c}$ are the position vectors of the points $A(1,3,0), B(2,5,0), C(4,2,0)$ respectively and $\bar{c}=t_{1} \bar{a}+t_{2} \bar{b}$,then the value of $t_{1} t_{2}$ is:

If $\bar{c} = 5\bar{a} + 6\bar{b}$ and $3\bar{c} = \bar{a} - 4\bar{b}$,then:

If $\bar{a} = (x + 2y - 3) \bar{i} + (2x - y + 3) \bar{j}$ and $\bar{b} = (3x - 2y) \bar{i} + (x - y + 1) \bar{j}$ are two vectors such that $\bar{a} = 2 \bar{b}$, then $y - 5x =$