a) Describe the shape of the intersection of the plane z=-3 and the plane y=z in three-space.
b) list all the possibilities for the intersection of a line and a plane, and draw an example of each.
help is much appreciated
to a):
If two planes are not parallel then they will intersect in a straight line.
The given planes are not parallel therefore you'll get a straight line as the intersection between them:
$\displaystyle P_1: (0,0,1)\cdot (x,y,z) = -3$ that means: $\displaystyle z = -3
$
$\displaystyle P_": (0,1,-1) \cdot (x,y,z)=0$ that means: $\displaystyle y-z=0$
Calculate:
$\displaystyle P_1 \cap P_2 = \left\{\begin{array}{lr}x = &t \\y =& -3+0\cdot t \\z=& -3+0\cdot t\end{array}\right.~\implies~(x,y,z)=(0,-3,-3)+t\cdot(1,0,0)$
to b): Let l denote the line and p the plane then
1. $\displaystyle l\cap p = \{P\}\ if\ l \nparallel p$
2. $\displaystyle l\cap p = \emptyset\ if\ l \parallel p$
3. $\displaystyle l\cap p = l\ if\ l \in p$