If $$\vec{a}, \vec{b}, \vec{c}$$ are three non-zero vectors, no two of them are collinear, $$\vec{a}+2 \vec{b}$$ is collinear with $$\vec{c}, \vec{b}+3 \vec{c}$$ is collinear with $$\vec{a}$$, then $$\vec{a}+2 \vec{b}$$ is

If

then $$|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}| \text { is }$$

If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are three vectors, $$|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=1, |\bar{b} \times \bar{c}|=\sqrt{15}$$ and $$\bar{b}=2 \bar{c}+\lambda \bar{a}$$, then the value of $$\lambda$$ is

Two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ are given by $$\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}$$ and $$\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$$. The side $$\mathrm{AD}$$ is rotated by an acute angle $$\alpha$$ in the plane of parallelogram so that $$\mathrm{AD}$$ becomes $$\mathrm{AD}^{\prime}$$. If $$\mathrm{AD}^{\prime}$$ makes a right angle with side AB, then the cosine of the angle $$\alpha$$ is given by