Problems

In Problems 1-14, find a vector equation for the given line.

1             The line through P(3, - 1) with direction vector D = -i + j.

2            The line through P(0,0) with direction vector D = i + 2j.

3            The line with parametric equations x = 3 - 2t, y = 4 + 5t.

4            The line with parametric equations x = 4t, y = 1 + t.

5            The line through the points P(1, 4) and Q(2, -1).

6            The line through the points P(5, 5) and Q(-6, 6).

7            The vertical line through P(2, 5).

8            The horizontal line through P(4, 1).

9            The line y = 2 + 5x.

10            The line x + y = 3.

11             The line y = 3.

12            The line x = y.

13            The line through P(6, 5) with slope -3.

14            The line through P(l, 2) with slope 4.

15            Find a scalar equation for the line X = 3i - 4j + f(i - 2j).

16             Find a scalar equation for the line X = 2i + t(-i + 4j).

17            Find a scalar equation for the line X = i + 3j + 4ti.

18            Find a scalar equation for the line with parametric equations x = 3 - 4t, y = 1 + 2t.

In Problems 19-24, determine whether the given three points are on a line.

25            Find the midpoint of the line AB where A = (2, 5), B = (-6, 1).

26            Find the midpoint of the line AB where A = (-1, -4), B = (9, 16).

27            Find the midpoint of the line AB where A = (5, 10), B = (-1, 10).

28            Find the point of intersection of the diagonals of the parallelogram A(1, 4), B(6, 4), C(6, 6), 0(1, 6).

29            Find the point of intersection of the diagonals of the parallelogram A(2, 0), B(5,1), C(6, 6), D(3, 5).

30            Find the point of intersection of the lines from the vertices to the midpoints of the opposite sides of the triangle ABC, where A = (1,4), B = (2, - 1), C = (6, 3).

31             Prove that the slope of a line with direction vector D = d₁i + d₂j is m = d₂/d₁ (vertical if d₁= 0).

32            Prove that if the diagonals of a four-sided figure bisect each other then the figure is a parallelogram. (Converse of Example 7.)

33            Prove that if the opposite sides of a four-sided figure are scalar multiples of each other then the figure is a parallelogram (i.e., the opposite sides are equal as vectors).

34            Let ABC be a triangle and let A₁,B₁, C₁ be the midpoints of the sides opposite A, B, C respectively. Show that the line AA₁ bisects the line B₁C₁.

35            Show that the midpoints of the sides of any four-sided figure are the vertices of a parallelogram.

36            Given a triangle ABC, let D be the midpoint of AB and E the midpoint of AC. Show that DE is parallel to BC and DE has half the length of BC. Hint: Show that E - D = ½(C - B).