If the points with position vectors $\alpha \hat{i} + 10 \hat{j} + 13 \hat{k}$,$6 \hat{i} + 11 \hat{j} + 11 \hat{k}$,and $\frac{9}{2} \hat{i} + \beta \hat{j} - 8 \hat{k}$ are collinear,then $(19 \alpha - 6 \beta)^2$ is equal to $...........$.

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$,then $\vec{a} + 2\vec{b} + 6\vec{c} = \dots$

If $2\vec{a} - 3\vec{b}$,$\vec{b}$,and $\vec{a} - \vec{b}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then they are:

If $\bar{x}$ is a non-zero vector and $k > 0, k \neq 1$,then $\frac{-k \bar{x}}{|\bar{x}|}$ is $.......$ .

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 $\lambda - \mu =$

If the position vectors of the vertices of a triangle are $6i + 4j + 5k$,$4i + 5j + 6k$,and $5i + 6j + 4k$,then the triangle is