Let \(\mu_{r 1}=2\) in region 1, defined by \(2 x+3 y-4 z>1\), while \(\mu_{r 2}=5\) in region 2 where \(2 x+3 y-4 z<1\). In region 1, \(\mathbf{H}_{1}=50 \mathbf{a}_{x}-30 \mathbf{a}_{y}+\) \(20 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m} .\) Find \((a) \mathbf{H}_{N 1} ;(b) \mathbf{H}_{t 1} ;(c) \mathbf{H}_{22} ;(d) \mathbf{H}_{N 2} ;(e) \theta_{1}\), the angle between \(\mathbf{H}_{1}\) and \(\mathbf{a}_{N 21} ;(f) \theta_{2}\), the angle between \(\mathbf{H}_{2}\) and \(\mathbf{a}_{N 21}\).

In this exercise, we are given the magnetic field in region 1, \(\mathbf{H_1}\), and need to determine several components and angles in regions 1 and 2. Follow these steps to find the required values: 1. Find the normal vector to the boundary surface (\(\mathbf{a}_{N21}= 2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z\)) and its magnitude (\(| \mathbf{a}_{N21} |= \sqrt{29}\)). 2. Find the normal component of \(\mathbf{H_1}\) (\(\mathbf{H}_{N1}\)) and the tangential component (\(\mathbf{H}_{t1}\)). 3. Using the boundary conditions, find \(|H_{N2}|\) and \(\mathbf{H}_{22}\). 4. Calculate the angles \(\theta_1\) and \(\theta_2\) between the magnetic fields and the normal vector using the dot product. By taking these steps, we can find the normal and tangential components of the magnetic field in each region and the angles between the fields and the normal vector.