If $$|\bar{a} \times \bar{b}|^2+(\bar{a} \cdot \bar{b})^2=144$$ and $$|\bar{a}|=4$$, then $$|\bar{b}|=$$

The distance between parallel lines

$$\begin{aligned} & \bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-2 \hat{k}) \text { and } \\ & \bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-2 \hat{k}) \text { is } \end{aligned}$$

The vertices of triangle $$\mathrm{ABC}$$ are $$\mathrm{A} \equiv(3,0,0) ; \mathrm{B} \equiv(0,0,4) ; \mathrm{C} \equiv(0,5,4)$$. Find the position vector of the point in which the bisector of angle A meets B C is

In a quadrilateral PQRS, M and N are mid-points of the sides PQ and RS respectively. If $$\overline {PS} + \overline {QR} = t\overline {MN} $$, then t =