Final answer:
The voltage (V) along the path from point A to B in a uniform electric field is determined by integrating the electric field along the displacement from A to B, which can be done using the dot product method. For the given vectors, the calculation shows that there is no voltage difference along this path since the result of the dot product is zero.
Step-by-step explanation:
The question is asking to calculate the voltage (V) along the path from point A to point B in a region with a uniform electric field. The electric field is given by the vector <-3, 4, 6> V/m, and the coordinates for points A and B are <1, 1, 0> m and <3, 4, -1> m, respectively. The voltage along the path can be determined by integrating the electric field along the displacement from A to B. One way to calculate this is to use the dot product of the electric field vector and the displacement vector from A to B. The displacement vector is found by subtracting the coordinates of A from those of B (B - A), which yields <2, 3, -1>. The dot product of the electric field and the displacement vector gives us the voltage along the path.
The dot product is calculated by multiplying corresponding components of the two vectors and then summing the results: V = -3*2 + 4*3 + 6*(-1) = -6 + 12 - 6 = 0 V. So, there is no voltage difference along the path between A and B. This result is consistent with the characteristics of a uniform electric field, as a uniform electric field implies that the potential difference between any two points that lie along an equipotential line is zero.