Question

A. i. On graph paper, draw the lines L₁:r= t(0, 1), IER, and L₂:r = p(1, 0), pER. Make sure that you clearly show a direction vector for each line. ii. Describe geometrically what each of the two equations represent. iii. Give a vector equation and corresponding parametric equations for each of the following: • the line parallel to the x-axis, passing through P(2, 4) • the line parallel to the y-axis, passing through Q(-2, −1) iv. Sketch L3: x = −3, y = 1 + s, sER, and L4: x = 4 + t, y = 1, tER, using your own axes. v. By examining parametric equations of a line, how is it possible to determine by inspection whether the line is parallel to either the x-axis or y-axis? vi. Write an equation of a line in both vector and parametric form that is parallel to the x-axis. vii. Write an equation of a line in both vector and parametric form that is parallel to the y-axis. Note: ER means it belong to all real numbers and the r is supposed to have an arrow above it referring to a vector

Answer 1

To graph lines, draw the lines L₁:r= t(0, 1) and L₂:r = p(1, 0) on graph paper with their direction vectors. Geometrically, L₁ represents a line parallel to the y-axis and L₂ represents a line parallel to the x-axis. For lines parallel to the x-axis or y-axis passing through specific points, use vector and parametric equations. For lines parallel to the x-axis, the equations will be r = a + s(1, 0) and x = a + s, y = b. For lines parallel to the y-axis, the equations will be r = a + t(0, 1) and x = b, y = a + t.

Step-by-step explanation:

Graphing Lines

1. Draw the lines L₁:r= t(0, 1) and L₂:r = p(1, 0) on graph paper, making sure to show a direction vector for each line.
2. Geometrically, the equation L₁:r= t(0, 1) represents a line parallel to the y-axis, and the equation L₂:r = p(1, 0) represents a line parallel to the x-axis.
3. a) Line parallel to the x-axis passing through P(2, 4): Vector equation: r = (2, 4) + s(1, 0), Parametric equations: x = 2 + s, y = 4. b) Line parallel to the y-axis passing through Q(-2, -1): Vector equation: r = (-2, -1) + t(0, 1), Parametric equations: x = -2, y = -1 + t.
4. Sketch L₃: x = -3, y = 1 + s and L₄: x = 4 + t, y = 1 on your own axes.
5. By examining the parametric equations of a line, if the line only has one variable, it is parallel to either the x-axis or y-axis.
6. Equation of a line parallel to the x-axis: Vector equation: r = a + s(1, 0), Parametric equations: x = a + s, y = b. (a is any real number)
7. Equation of a line parallel to the y-axis: Vector equation: r = a + t(0, 1), Parametric equations: x = b, y = a + t. (a is any real number)