If $\bar{a}$ and $\bar{b}$ are vectors such that $|\bar{a}|=3$,$|\bar{b}|=2$ and $\bar{a} \cdot \bar{b}=5$,then $|\bar{a}-\bar{b}|=$

If $\hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+2 \hat{j}+\hat{k}$ are sides of a parallelogram,then a unit vector parallel to one of the diagonals of the parallelogram is

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k},$ find a unit vector parallel to the vector $2 \vec{a}-\vec{b}+3 \vec{c}.$

If $p = (7, -2, 3)$ and $q = (3, 1, 5)$,then the magnitude of $p - 2q$ is $......$.

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 $16(3a-1)^2=$

If the vectors $\vec{a} = \lambda \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = \sqrt{\lambda} \hat{i} + \sqrt{13} \hat{j}$ have the same magnitude,then the value of $\lambda$ is: