If $p = 7i - 2j + 3k$ and $q = 3i + j + 5k$,then $|p - 2q| = \dots$

If the origin is the orthocenter of an equilateral triangle whose vertices are represented by the position vectors $\vec{a}, \vec{b}, \vec{c}$,then which of the following is true?

If the position vectors of the points $A, B, C, D$ given by $\hat{i}+2 \hat{j}+3 \hat{k}, 2 \hat{i}-\hat{j}+2 \hat{k}$, $\frac{1}{4}(7 \hat{i}+15 \hat{j}+15 \hat{k})$ and $\frac{1}{3}[7 \hat{i}+2 \hat{j}+(5+3 a) \hat{k}]$ respectively are such that $|AC|=|BD|$, then $16(3a-1)^2=$

In a regular hexagon $ABCDEF$,$AD + EB + FC = (3\lambda - 8) AB$. Then $\lambda =$

The direction cosines of the vector $\hat{i}-2\hat{j}+3\hat{k}$ are . . . . . . .

If $(\alpha, \beta, \gamma)$ is a triad of real numbers satisfying $\hat{i}-2 \hat{j}+5 \hat{k}=\alpha(\hat{i}+\hat{j}+\hat{k})+\beta(\hat{i}+2 \hat{j}+3 \hat{k})+\gamma(2 \hat{i}-\hat{j}+\hat{k}),$ then $\alpha^2-\beta^2+\gamma^2=$