Read the following two definitions carefully, and then answer the question below. Given a plane P, a line L and a vector v in R³, we define the following new plane and new line: • P+ {v} = {x+ v | x €...

answer 2.1 and 2.2

Extracted text: Read the following two definitions carefully, and then answer the question below. Given a plane P, a line L and a vector v in R³, we define the following new plane and new line: • P+ {v} = {x+ v | x € P} and •L+ {v} = {x+v | x € L}. For example, if v = [0 0 1]† and L is the x-axis, then L+ {v} is the line parallel to L sitting one unit directly “above" L (in the z direction). In other words, the line L has been "shifted" by the vector v. This is a special case of much more general definition we could make: if A and B are any sets of vectors (say in R3) or numbers or matrices, or really anything that we can 'add' we can let A+ B be the set {a +b | a € A,b € B}. 1 We say that a plane P or line L is original if it goes through the origin. Now, if P is an original plane, we define a plane P' to be a a shift of P if P' = P+ {v} for some non-zero v. In this case, we say that we shifted P by v to get P' or that P' is P shifted by v. (We can make analogous definitions for lines: e.g. if L is an original line, we define a line L' to be a shift of L if L' = L+ {v} for some non-zero v.) 2.1 Explain why if P is an original plane, and w and x are in P, w+x is also in P. (i.e. the sum of two vectors in an original plane is always a vector in that same plane.) 2.2 Explain why the following statement is false: "If P' is a shift of some (original) plane P in R3, say P' = P+ {v}, then for any w and x in P', w +x is in P'." (This shows that we can't simply add two vectors in a shift of a plane and have the result be a vector in that shifted plane.) 2.3 Suppose we "fix" the statement in the the previous part by changing it to the following: "If P' is a shift of some (original) plane P in R°, say P' = P+ {v}, then any w and x in P', w + x – v is in P'." Explain why this statement is true. for Hint: Is there a way to add the vectors "inside P" and then shift the result to P'? Alternately, it could help to think of V (the point for v) as the "origin" for P'. Definition Definition