Question

how to determine if two lines are perpendicular based on their slopes

Answer 1

To determine if two lines are perpendicular based on their slopes, you can use the following rule:

1. **Identify the slopes**: Let the slopes of the two lines be represented as \( m_1 \) and \( m_2 \).

2. **Use the perpendicularity condition**: Two lines are perpendicular if the product of their slopes is -1. In mathematical terms, this can be expressed as:
\[
m_1 \times m_2 = -1
\]

3. **Interpret the result**: If the condition is satisfied (i.e., the product equals -1), then the lines are perpendicular. If the product does not equal -1, then the lines are not perpendicular.

### Example
- If line 1 has a slope of \( m_1 = 2 \) and line 2 has a slope of \( m_2 = -\frac{1}{2} \):
\[
m_1 \times m_2 = 2 \times \left(-\frac{1}{2}\right) = -1
\]
Since the product is -1, the lines are perpendicular.

- If line 1 has a slope of \( m_1 = 3 \) and line 2 has a slope of \( m_2 = 2 \):
\[
m_1 \times m_2 = 3 \times 2 = 6
\]
Since the product is not -1, the lines are not perpendicular.

### Summary
Simply calculate the product of the slopes of the two lines. If the result is -1, the lines are perpendicular.