If $\vec{a} = -4\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 12\hat{i} - 6\hat{j} + 15\hat{k}$,then the vectors $\vec{a}$ and $\vec{b}$ are $.......$

$A$ point $C$ with position vector $\frac{3 \bar{a}+4 \bar{b}-5 \bar{c}}{3}$ (where $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar vectors) divides the line segment joining $A$ and $B$ in the ratio $2:1$. If the position vector of $A$ is $\bar{a}-2 \bar{b}+3 \bar{c}$,then find the position vector of $B$.

The vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-2 \hat{j}+5 \hat{k}$ and $\vec{b}=-2 \hat{i}+5 \hat{j}-3 \hat{k}$ is

If three points $A$,$B$,and $C$ have position vectors $(1, x, 3)$,$(3, 4, 7)$,and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y)$ is

If $ABCDEF$ is a regular hexagon and $\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + \overrightarrow{AF} = \lambda \overrightarrow{AD}$,then $\lambda = $

If $PQ + QR = (2\lambda^2 - 5)RP$,then $\lambda$ is equal to