Vector Algebra

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