1 Can two lines be drawn with the given intersection Support

1. Can two lines be drawn with the given intersection? Support your answer with a diagram, an example, a real-life situation, or an explanation. Be sure to answer the question for all four intersections given.

a. a point b. a Line c. a intersection

I have answered for all the above.

d. A region. I am struggling with this. How the region will form. Kindly help me out.

2. Find the possible intersections for the graphs of more than two linear inequalities. support each possibility with a diagram.

Thanks in advance.

Solution

a) a point: yes - consider two lines similar to the letter X (different slopes, they will meet)

b) a line: yes - two lines can intersect in a line if they are the same line

c) a region: no - the dimension of a region I presume is similar to that of a plane which is 2, but a line which has dimension 1 cannot intersect with another line and have a dimension greater than the minimum of both the two spaces, which is 1. that is if U and V are lines, then dim(U) = dim(V) = 1, so dim(U intersection V) is either less than or equal to 1

d) no intersection: yes - parallel lines, or skew lines (in R3), such as //. Skew lines you can think about if you put out your hands in a cross shape but they are not touching, and they are not parallel. It\'s hard to describe but you can sort of intuitively think of it in R3