$$ \text { If } \hat{\imath}+\hat{\jmath}-\hat{k} \quad \&~ 2 \hat{\imath}-3 \hat{\jmath}+\hat{k} \text { are adjacent sides of a parallelogram, then length of its diagonals are } $$

Find the value of '$$b$$' such that the scalar product of the vector $$\hat{\imath}+\hat{\jmath}+\hat{k}$$ with the unit vector parallel to the sum of the vectors $$2 \hat{\imath}+4 \hat{\jmath}-5 \hat{k}$$ and $$b \hat{\imath}+2 \hat{\jmath}+3 \hat{k}$$ is unity

The vector $$(\vec{r})$$ whose magnitude is $$3 \sqrt{2}$$ units which makes an angle of $$\frac{\pi}{4}$$ and $$\frac{\pi}{2}$$ with $$y$$ and $$z$$- axis respectively is

$$ \text { If }|\vec{a} \times \vec{b}|^2+|\vec{a} \cdot \vec{b}|^2=144 ~\&~|\vec{a}|=4 \text { then }|\vec{b}|= $$