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

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

Let a, b, c be three vector such that $$a \neq 0$$ and $$\vec{a} \times \vec{b}=2 \vec{a} \times \vec{c},|a|=|c|=1,|b|=4$$ and $$|\vec{b} \times \vec{c}|=\sqrt{15}$$. If $$\vec{b}-2 \vec{c}=\lambda \vec{a}$$ then $$\lambda$$ equals to

$$ \text { The angle between } \hat{\imath}-\hat{\jmath} ~\&~ \hat{\jmath}-\hat{k} \text { is } $$