Let l1, l2, and lz be nonconcurrent lines. Prove that there exist three points, exactly one each from each of l1, l2, and l3, that are collinear.

Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.

If l,, l2, and l3 are nonconcurrent lines, prove that there exist three points. Exactly one each from each of l1, l2, and l3 must be collinear.

Find k so that the points P(0, 1, 1), Q(-1, 1, 3), and R(-3, 1, k) are on the same line (collinear).

Prove that three points P(5,-4), Q(-3,8) and R(1,2) on the rectangular coordinate plane are collinear.

If (x1, yı) and (x2, y2) are distinct points in the plane, show that y 1 1 =0 X1 x2 y2 is the equation of the line through (x1, yı) and (x2, y2). Use this result to develop a test for collinearity of three yı 1 points.

Let R be the parallelogram enclosed by the lines x + 3y = 0, x + 3y = 2, x + y = 1 and x + y = 4. Use the transformation u = x + 3y and v = x + y to evaluate x + 3y dA. R

Consider the line segments (1,2) and (3,4). Prove that the lines are (i) non- collinear (ii) non-overlapping, (iii) Intersecting.

Let R denote the rectangle with corners (-1,-2), (4, -2), (4,1), and (-1, 1), and let P denote the path going clockwise around this rect- angle, starting and ending at the point (-1,1). Compute pF. dr, where F(x, y) = (y). Let R denote the rectangle with corners (1, 2), (1,8), (4,2), and (4,8), and let P denote the path going counterclockwise around this rectan- gle, starting and ending at the point (1,2). Compute ſpF · dř, where F(x, y) = (2²+y).

Q.2) The horizontal line h in space placed in position that the coordinates of its two ends are D(5,5,2) and E(?,?,?). If the point E is located on the straight line m[A(1,3,1),B(6,2,3)], draw the two projections of of the horizontal line h.