**Geometric Structures: Practice 4: 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 following determinants by using the number in the (1,1) position to "clear the first column," that is, place the number zero in the (2,1) and (3,1) positions.

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

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

b. (4, 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. x_{1} + 3x_{2} - x_{3} = 0 and x_{1} - 2x_{2} + 3x_{3} = 0

b. x_{1} + 2x_{2} - 2x_{3} = 0 and x_{1} - 1x_{2} + 1x_{3} = 0

c. 2x_{1} - 3x_{2} - 2x_{3} = 0 and x_{1} - x_{2} + x_{3} = 0

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

For example:

x_{1} + 3x_{2} - x_{3} = 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.

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