3a If L is a line whose equation is ar by c for some a b c e

3a. If L is a line whose equation is ar by c for some a, b, c e Re, and if M is a line parallel to L, what can you say about the equation for M? Justify your answer b. Prove: L and M are parallel lines in R2 if and only if there is a translation map Tor, s) R2 R2 mapping L onto M

Solution

3. a)

Let L be the given line whose form is ax+by=c for some a,b,c in R.

Then a line M is parellel to the line L if the co-efficients of line M are proportional to a,b,c say na,nb,nc for some n.

which means that the lines are parellel to each other if the direction cosines are same .

3.b)

Consider L and M are parellel lines in R^2.     which means that there is a mapping between each and every points of the lines. The points in L and M are separated by a constant diffrence. so there is a mapping which maps a single point of one line on the other line , hence the Mapping is onto.

Conversly,

Suppose the mapping between the lines is onto which means there is only one image for each point which leads to linear correspondence between the points , Hence those two lines are parellel,

Hence the proof