If $$[(\bar{a}+2 \bar{b}+3 \bar{c}) \times(\bar{b}+2 \bar{c}+3 \bar{a})] \cdot(\bar{c}+2 \bar{a}+3 \bar{b})=54$$ then the value of $$\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]$$ is

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is $$3 \mathrm{~mm}$$ and 1 hour later it reduces to $$2 \mathrm{~mm}$$, then the expression for the radius $$R$$ of the raindrop at any time $$t$$ is

If the Cartesian equation of a line is $$6 x-2=3 y+1=2 z-2$$, then the vector equation of the line is

The volume of parallelopiped, whose coterminous edges are given by $$\overline{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \vec{v}=\hat{i}+\hat{j}+3 \hat{k}, \bar{w}=2 \hat{i}+\hat{j}+\hat{k}$$ is 1 cu. units. If $$\theta$$ is the angle between $$\bar{u}$$ and $$\bar{w}$$, then the value of $$\cos \theta$$ is