ECH3264 Transport Phenomena I Spring'97

Supplemental Problems

1. Decompose the vector into vectors parallel and perpendicular to the vector .

2 A plane is determined by the point ) on it, and the vector
normal to it.

(i) Show that the requirement that the vector from to an arbitrary point P(x,y,z) in the plane be perpendicular to , determines the equation of the plane in the form:

this may also be written in the form:

where d = ax + by + cz

(ii) Show that the shortest distance from the point P ) to the plane
ax + by + cz = d is given by:

(iii) Show that the shortest distance from a point to the line joining the points Q and Q is given by:

3. The temperature of points in space is given by . A mosquito located at (1,1,2) desires to fly in a direction so that it will get cool as soon as possible. In what direction should it move?

4. Sketch the vector field f=xi+yj+zk and calculate its curl. In terms of the paddle-wheel interpretation of the curl, explain why curl f is identically zero in this case.