Strain at a point is such that \(\varepsilon_{x x}=\varepsilon_{y y}=0, \varepsilon_{z z}=-0.001, \varepsilon_{x y}=0.006\), and \(\varepsilon_{x z}=\varepsilon_{y z}=0\). Note: You need not solve the eigenvalue problem for this question.

a. Show that \(\mathbf{n}^{1}=\mathbf{i}+\mathbf{j}\) and \(\mathbf{n}^{2}=-\mathbf{i}+\mathbf{j}\) are principal directions of strain at this point.

b. What is the third principal direction?

c. Compute the three principal strains.