In physics, a displacement vector is crucial as it helps us to understand the movement from one point to another in space. It is defined as the difference between the final and initial position vectors. To get the displacement vector between two points, we subtract the initial position's coordinates from the final position's coordinates.
In the given exercise, for paths (a) and (b), we have:
- Point P: (1, 2, 3)
- Point Q: (2, 1, 4)
Therefore, the displacement vectors were calculated as:
- From P to Q: \( \Delta \mathbf{d}_a = Q - P = (1, -1, 1) \)
- Normalized and scaled by 2 mm: \( \Delta \mathbf{d}_a = \frac{2}{\sqrt{3}}(1, -1, 1) \)
- From Q to P: \( \Delta \mathbf{d}_b = P - Q = (-1, 1, -1) \)
- Normalized and scaled by 2 mm: \( \Delta \mathbf{d}_b = -\frac{2}{\sqrt{3}}(1, -1, 1) \)
This step-by-step subtraction and normalization help clarify how movement occurs within an electromagnetic field.