The range from \( t = 0 \) to \( t = 1 \) is used because we are specifically dealing with a line **segment** \( AB \), not the entire line that extends infinitely in both directions.

### Explanation:
1. **Parametric Representation:**
   - The parametric form of a line through points \( A \) and \( B \) can be expressed as:
     \[
     \mathbf{r}(t) = \mathbf{A} + t(\mathbf{B} - \mathbf{A})
     \]
     where:
     - \( \mathbf{A} \) is the position vector of point \( A(4,1,2) \).
     - \( \mathbf{B} \) is the position vector of point \( B(3,0,2) \).
     - \( t \) is a parameter that controls the position along the line.

2. **Range Interpretation:**
   - \( t = 0 \) gives the point \( A(4,1,2) \).
   - \( t = 1 \) gives the point \( B(3,0,2) \).
   - For \( 0 < t < 1 \), the point lies somewhere between \( A \) and \( B \) on the segment.
   - For \( t < 0 \), the point lies on the extension of the line beyond \( A \).
   - For \( t > 1 \), the point lies on the extension of the line beyond \( B \).

Thus, restricting \( t \) to the interval [0, 1] ensures that the points we consider are only those on the segment between \( A \) and \( B \).

### Why is this important?
When determining if the segment intersects the plane, we only care about the portion of the line that exists between \( A \) and \( B \). If the value of \( t \) corresponding to an intersection point is outside the range [0, 1], it means the intersection occurs outside the segment, either before point \( A \) or after point \( B \).

Would you like more details, or do you have any other questions?

Here are five related questions to expand on this concept:

1. How do you generalize the parametric form for a line passing through two points in 3D space?
2. What does the parameter \( t \) represent geometrically when it lies outside the range [0, 1]?
3. How can you determine the length of the line segment \( AB \) using vector algebra?
4. What is the difference between a line and a line segment in terms of their geometric properties?
5. How would you determine if the entire line (not just the segment) intersects with the plane?

**Tip:** When working with line segments, always check if your solution lies within the defined range of the parameter to ensure you're considering the correct portion of the line.

Explore why the range from t = 0 to t = 1 is used in parametric equations to represent line segments in 3D space. Learn how this range ensures accurate segment representation and intersects with planes correctly.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation