Question

1. what are the possible relationships between lines (ex: parallel)? for each, explain what it is or draw a picture. please list at least 4 relationships for 2d, and one for 1d/3d. 4. what do the slopes of lines tell us about their relationships to one another? (i.e. for each of the relationships listed in question 3, what about their slopes indicate this?)

Explanation:

3.

• Parallel: In 2 - D, two lines are parallel if they never intersect and have the same slope. For example, the lines \(y = 2x+1\) and \(y = 2x + 3\).
• Intersecting: In 2 - D, two lines that cross at a single point. For example, \(y=x\) and \(y = 2 - x\) intersect at \((1,1)\).
• Perpendicular: In 2 - D, two lines are perpendicular if the product of their slopes is - 1. For example, \(y=x\) and \(y=-x\).
• Co - incident: In 2 - D, two lines are co - incident if they are the same line, e.g., \(y = 2x+1\) and \(2y=4x + 2\). In 3 - D, skew lines are lines that are not parallel and do not intersect. They exist in different non - parallel planes.

4.

• Parallel lines: Have equal slopes. If \(m_1\) and \(m_2\) are the slopes of two lines, for parallel lines \(m_1=m_2\).
• Intersecting lines: Have different slopes (\(m_1

eq m_2\)).

• Perpendicular lines: The product of their slopes is - 1, i.e., \(m_1\times m_2=-1\).
• Co - incident lines: Have the same slope and the same y - intercept.

Answer:

3.

• Parallel: Lines never intersect, same slope in 2 - D.
• Intersecting: Lines cross at a point in 2 - D.
• Perpendicular: Product of slopes is - 1 in 2 - D.
• Co - incident: Same line in 2 - D. Skew lines in 3 - D (not parallel, don't intersect).

4.

• Parallel: \(m_1 = m_2\).
• Intersecting: \(m_1

eq m_2\).

• Perpendicular: \(m_1\times m_2=-1\).
• Co - incident: Same slope and y - intercept.