Problem 28. The point   Q(5,-2)   is the vertex of the angle. We call the given points on the two sides of the angle   P = (4,-1)   and   R = (0,0).   The line joining   P   and   Q   is given in normal form as   (1,1),X = 3,   or equivalently   x + y = 3.   The line joining   Q   and   R   is given in normal form as   (2,5),X = 0,   or equivalently 2x + 5y = 0.

By definition, the interior of the angle consists of all points that are on the same side of   PQ   as   R   and also on the same side of   QR   as   P.   We calculate:   (1,1),R = (1,1),(0,0) = 0 < 3.   This shows that the points on the same side of   PQ   as   R   are characterized by the inequality (1,1),X < 3,   or equivalently   x + y < 3.
 
A similar calculation shows that   (2,5),P = (2,5),(4,-1) = 3 > 0.   This shows that the points on the same side of   QR   as   P   are characterized by the inequality   (2,5),X > 0,   or equivalently   2x + 5y > 0.   Therefore, the interior of the angle is characterized by the following system of inequalities:
         x + y < 3 ;
       2x + 5y > 0 .
 

Problem 53. One side of the angle has direction indicator   (3,4) - (2,-1) = (1,5).   The length of this vector is   26^1/2   [square root of 26 . . .],   so that we obtain the unit direction indicator   U = ¹/_26^1/2 (1,5). The other side of the angle has direction indicator   (4,5) - (2,-1) = (2,6), which has length = 40^1/2 = 2·10^1/2.   Therefore, the unit direction indicator of this side is   V = ¹/_40^1/2(2,6),   or   V = ¹/_10^1/2(1,3).   To find the angular measure, we first calculate the inner product:

U,V =

1      261/2

·

1      101/2

(1,5),(1,3) =

1      2601/2

(1,5),(1,3) =

16     2601/2

=

8

The angular measure is then given as the following integral:

1   8/

ds      (1 - s2)1/2

Problem 69. Let the equation of  l  be given in normal form as  A,X = c,  and let the equation of  k  be  A,X = b.   (We can use the same vector  A  in both equations, since the lines are parallel to each other. According the formula in Theorem 23, the distance to  l  from a point  Y  is:

|c - A,Y |       ||A||

But if  Y  lies on  k,  then  A,Y = b. In this case, the formula for the distance can be simplified to   ^|c - b| / _||A||.   This does not depend on the choice of a point on k.

Problem 70.   {Not assigned}   Let the equations of the lines  k  and  l  be as in Problem 69. By the result of that problem, the distance to  l  from a point on  k  is  ^|c - b| / _||A||,  while the distance to  k  from a point on  l  is  ^|b - c| / _||A||.  Since the absolute value of any real number is equal to the absolute value of its negative, we conclude that these two numbers are equal.