Final answer:
The component functions of a vector field F in R² that is everywhere normal to the line y = 5 is given by option D, F = (0, y), since it is the only option that provides a consistent vertical vector field normal to the horizontal line.
Step-by-step explanation:
To specify the component functions of a vector field Φ in R² that is normal to the line y = 5, we can analyze the given options. The vector field should have the property that its direction at any point is perpendicular to the line y = 5. In other words, if you think of the line y = 5 as being horizontal, the vector field at any point on this line should be vertical to ensure that it is normal (at a 90 degrees angle) to the line.
Considering the options presented:
- A. F = (-y, x) has components that depend on both x and y and does not consistently yield a vertical vector.
- B. F = (x, y) does not give a vector always normal to the line y = 5, as this would imply a vector changing its direction with changing x and y.
- C. F = (x, 0) is not always normal since its direction is purely horizontal, along the x-axis.
- D. F = (0, y) correctly represents a vector field that is always vertical since it has no x-component and its magnitude varies with y, which is the correct answer.
By analyzing the component functions of a vector field normal to a specified line, one can deduce that option D provides a vector field that is everywhere normal to the line y = 5.
The general idea in such a case is that if a vector is normal to a surface or line, its components will not have any component along the direction of the line itself. Therefore, since the line y = 5 is horizontal, the vector field normal to it will only have a y-component.