The stress matrix at a point \(P\) is given below. The direction cosines of the normal \(\mathbf{n}\) to a plane that passes through \(P\) have the ratio \(n_{x}: n_{y}: n_{z}=3: 4: 12\). Determine:

(a) the traction vector \(\mathbf{T}^{(\mathbf{n})}\);

(b) the magnitude \(T\) of \(\mathbf{T}^{(\mathbf{n})}\);

(c) the normal stress \(\sigma_{n}\);

(d) the shear stress \(\tau_{n}\); and

(e) the angle between \(\mathbf{T}^{(\mathbf{n})}\) and \(\mathbf{n}\). Hint: Use \(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}=1\).

[13 13 0 [6]=13 26-13 0-13-39