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

The value of $$x$$, for which $$\sin \left(\cot ^{-1}(x)\right)=\cos \left(\tan ^{-1}(1+x)\right)$$, is

The unit vector which is orthogonal to the vector $$3 \hat{i}+2 \hat{j}+6 \hat{k}$$ and coplanar with the vectors $$2 \hat{i}+\hat{j}+\hat{k}$$ and $$\hat{i}+\hat{j}+\hat{k}$$ is

A ladder, 5 meters long, rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle between the ladder and the floor is decreasing at the rate of __________ radians/second when it's lower end is $$4 \mathrm{~m}$$ away from the wall.