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

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

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

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