Question

2. (From the distance test to vector stretches) Assume that A, B, C are points in the plane are on a line where B is in the middle, İ.e dist(AC)-dist(AB) + dist(BC). The goal of this exercise is to check that this is equivalent to the vector description! We will make some use of vectors and their intuition. În particular, if the coordinates of A, B, C are (zaJa), (Tb,Yb), (Te'%), we can translate them with a vector [u, v] by simply adding u to al the first coordinates and v to all the second coordinates, İ.e change for points A. B', C, with coordinates (za + u, ya + u), (Tp + u'yb + u), (a) Check that dist(A, B)dist(A', B') and similarly for the other two (b) Compute the vectors from A to B from A to C and from B to C. Do they depend on u and v? Thus we may translate B to be the point B with coordinates (0,0). This will not change results that only depend on the vectors and the distances above! To simplify notation, say that A and C has then coordinates (xi, yi) and (x2, /2) (c) Express x1.JI,T2,y2 in terms ofTa泓,ть,Ub,Те, ус (d) What can be said about the comparison between the signs of x1, 22 and yi,y2 (It is important that (0,0) is in the line!) This is an incredibly useful to simplify computations. We will use it rather often! From now on we work in the second setting, i.e with A, (0,0) and C, where the important fact is that c is in the middle (e) Write the equation dist(AC) -dist(AB) + dist(BC) in terms of the coordinates (f) Square the equations that you get and cancel terms. What do you get? This is equation is equivalent since the distances are positive another term on the other side. Check that both sides are again positive have to use that (a - b)22-2ab, just choose a and b wisely (g) After cancelation, you should have an equation that contains a square root on one side and (h) Square this new identity and factor it to obtain a number whose square is 0. Hint: You will (i) Explain why the result in h is equivalent to saying that the vectors of from B to A and from B to C are parallel, i.e one of them is a (negative!) stretch of the other!

Answer 1