Geometric Structures: Determinants, Points and Lines

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

1. Compute the value of the following determinants:

a.

b.

c.

2. Find the value of the following determinants by expanding using both the first row and the first column:

a.

b.

3. Find the value of the two determinants above by using the number in the (3,3) position to "clear the third column," that is, place the number zero in the (2,3) and (1,3) positions.

4. Find the line through the Euclidean points shown using determinants:

a. (2, 3) and (-1, 3)

b. (3, 6) and (-2, -3)

5. Find the line through the projective points shown using determinants:

a. (1, 2, 0) and (2, -1, 3)

b. (-1, 4, 1) and (2, 3, 1)

c. (3, 0, 1) and (-1, 3, 1)

6. Find the point where the projective lines meet:

a. x1 + 3x2 - x3 = 0 and x1 - 2x2 + 3x3 = 0

b. x1 + 2x2 - 2x3 = 0 and -1x1 - 2x2 + 3x3 = 0

c. x1 + 3x2 - x3 = 0 and x1 + 2x2 + 3x3 = 0

7. Write down the Euclidean lines that correspond to the above lines.

For example:

x1 + 3x2 - x3 = 0 corresponds to x + 3y = 1

(-2, 1) and (1, 0) are Euclidean points on the Euclidean line above and you can check that the corresponding points in the projective plane:

(-2, 1, 1) and (1, 0, 1) satisfy the equation of the the projective line.

8. Find the point where x + y = 2 and x - y = 4 meet and write down the real projective lines that correspond to these and find where these meet. Show that the answer corresponds to the Euclidean point.

9. Find the line in the Euclidean plane through (-1, 3) and which is parallel to 2x-3y = 4

10. In the real projective plane find three different lines which pass through each of the points below:

a. (1, 2, 1)

b. (-1, 3, 1/2)

c. (2, -1, 4)

11. Write down three other points whose coordinates represent the same point as the given one in the real projective plane:

a. (-1, -1, -1)

b. (2, 0, 0)

c. (-1, 0, 6)