1. Given loop vertices

The loop is formed by the points $A(0,0,0),\; B(5,0,0),\; C(5,5,0),\; D(0,5,0),\; E(0,5,5),\; F(0,0,5).$

So the closed polygon is $A \to B \to C \to D \to E \to F \to A$.

The magnetic field is uniform: $\vec B = 3\hat i + 4\hat k \quad (\text{tesla})$

We need the magnetic flux through this loop: $\Phi = \vec B \cdot \vec A$ where $\vec A$ is the vector area of the non-planar closed loop.

2. Use vector area of a closed polygon

For a closed loop with consecutive position vectors $\vec r_1, \vec r_2, \dots, \vec r_n$, $\vec A = \frac12 \sum_i (\vec r_i \times \vec r_{i+1})$ with $\vec r_{n+1}=\vec r_1$.

Let us compute this for the vertices in order: $A,B,C,D,E,F.$

Their position vectors are: $\vec r_A=(0,0,0),\; \vec r_B=(5,0,0),\; \vec r_C=(5,5,0),$ $\vec r_D=(0,5,0),\; \vec r_E=(0,5,5),\; \vec r_F=(0,0,5).$

Now calculate each cross product.

3. Compute the cross products

$\vec r_A \times \vec r_B = (0,0,0)$