By Aichinger E.
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Extra info for 2-affine complete algebras need not be affine complete
Find the area of the triangle whose vertices are A(1, - 3), B(2, 5), C(- 4, - 3). 14. Show that the line joining A(-1, 4, - 3) and B(5, -8, 6) meets the x axis. If the point of intersection is C, find the ratio AC/CB. 4. COORDINATE GEOMETRY IN A PLANE; STRAIGHT LINES AND THEIR GRADIENTS Throughout this section we shall confine our attention to a plane, so that only two coordinates are required to specify a point. The reader will recall that the vector equation of a straight line is the algebraic condition that must be satisfied by the position vector r of a point P if P is to lie on the line.
Later we shall meet loci that are defined by inequalities; the interior of a sphere would be such a locus. A straight line is completely specified if two points A, B of the line are given. Suppose that an origin 0 is taken and that the position vectors of A and B relative to 0 are a and b respectively. The equation of the line AB is the condition satisfied by the position vector r of a general point on AB. ) Now, since P lies on AB, or AB produced (Fig. 12), AP = AAB, where A is a number. For different points on the line, different values of A are taken.
Note again that the coordinates are given in the order x, y, z. Ex. 5. Draw a sketch to denote the approximate positions of the points: A(1, 0), B( -2, - 1), C(1, -3), D( -2, 0), E( -1, -2). Ex. 6. Write down the position vectors of the points A(1, - 2), B(3, 4). Deduce the coordinates of the mid-point of AB. Can you state a general rule for finding the coordinates of the mid-point of a line ? Ex. 7. Write down the position vectors of the points A(1, - 1), B(5, -5). Use the Section Formula to deduce the coordinates of the two points of trisection of AB.
2-affine complete algebras need not be affine complete by Aichinger E.