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 $$\overline{A B}=2 \hat{i}+10 \hat{j}+1 \hat{k}$$ 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

If the area of the triangle with vertices $$(1,2,0)$$, $$(1,0,2)$$ and $$(0, x, 1)$$ is $$\sqrt{6}$$ square units, then the value of $x$ is

Let $$\overline{\mathrm{A}}$$ be a vector parallel to line of intersection of planes $$P_1$$ and $$P_2$$ through origin. $$P_1$$ is parallel to the vectors $$2 \hat{j}+3 \hat{k}$$ and $$4 \hat{j}-3 \hat{k}$$ and $$P_2$$ is parallel to $$\hat{j}-\hat{k}$$ and $$3 \hat{i}+3 \hat{j}$$, then the angle between $$\bar{A}$$ and $$2 \hat{i}+\hat{j}-2 \hat{k}$$ is