Define the extended vector for each plane as follows: e1= [a1,b1,c1,d1] e2=[a2,b2,c2,d2] e3=[a3,b3,c3,d3]?

define the extended vector for each plane as follows: e1= [a1,b1,c1,d1] e2=[a2,b2,c2,d2] e3=[a3,b3,c3,d3]


use the extended (where necessary) and normal vectors of these planes to express the following geometric conditions.


a) 3 parallel, distinct planes


b) 2 parallel distinct planes intersected by another plane to form 2 parallel lines


c) 3 distinct planes forming a triangular prism, that is, no common points of intersection but intersecting in pairs to form 3 parallel lines.


e) 3 distinct planes intersecting in a unique point

Define the extended vector for each plane as follows: e1= [a1,b1,c1,d1] e2=[a2,b2,c2,d2] e3=[a3,b3,c3,d3]?
What are your "extended vectors"? I suppose, from the outcomes, that you are working in a 3-dimensional space.


a) The condition for two planes to be parallel is that their normal vectors are colinear (i.e. multiples of one another). The result is then easy (you just have to rule out the case where two are equal)


b) the "2 parallel lines" bit is automatic, as long as there is an intersection.


c) translate as "non-parallel", plus the condition that the three planes do not have a point in common.


d) same.


As far as parallel is concerned, normal vectors are sufficient; for the rest (intersection) you need more info (extended vectors probably, but i have never encountered the concept so far. It could be the coefficients of the equation?)