Question

If a plane contains two distinct points \(P_{1}\) and \(P_{2}\), show that it contains every point on the line through \(P_{1}\) and \(P_{2}\).

Short answer

Every point on the line between two points in a plane also lies on that plane.

Step-by-step solution

Understanding the Problem

We have two distinct points, \(P_1\) and \(P_2\), that lie on a plane. The problem asks us to show that any point on the line that connects \(P_1\) and \(P_2\) is also on the plane.

Parameterizing the Line

To describe the line passing through \(P_1\) and \(P_2\), we can use the vector equation of a line. The line can be parameterized as \(L(t) = P_1 + t(P_2 - P_1)\) for some parameter \(t\). Here \(L(t)\) describes any point on the line based on the value of \(t\).

Using Vector Addition

The expression \(L(t) = P_1 + t(P_2 - P_1)\) indicates that any point on the line is a linear combination of the vectors \(P_1\) and \(P_2\). Since planes are closed under linear combinations, any such combination will also be in the plane.

Conclusion

Since we have expressed every point on the line using the points \(P_1\) and \(P_2\) and since the plane includes all linear combinations of its contained points, every point \(L(t)\) on the line must also lie on the plane.

Key concepts

Vector Equation of a Line

In plane geometry, a vector equation of a line is a powerful tool that helps us describe the line in a mathematical way. Imagine this line extending through two distinct points, say \(P_1\) and \(P_2\), on a plane. We can write this line as \(\boldsymbol{L(t) = P_1 + t(P_2 - P_1)}\). Here, the term \(t\) represents a parameter, which is a real number.
This equation tells us two things:

• The vector \(P_1\) acts as an anchor point.
• The vector difference \((P_2 - P_1)\) gives direction to the line.

The magic here is that \(t\) scales the direction vector, moving the point along the line. When you change \(t\), you essentially walk along the line through these points. What this equation shows us is that every possible point on this line can be reached by modifying \(t\). The beauty of this approach is its simplicity and ability to handle infinite points on a line with just one formula.

Linear Combination

A linear combination is one of the foundational concepts in both algebra and geometry. It allows us to create new vectors from a given set by using scaling and addition. In our specific problem, the linear combination of \(P_1\) and \(P_2\) is shown in the vector equation \(L(t) = P_1 + t(P_2 - P_1)\). What does this mean?

• You take a part of \(P_1\), scaled by 1 (since \(P_1\) is the starting point).
• Then, you add a scaled version of the direction vector \((P_2 - P_1)\).

This process of scaling and adding vectors can result in any point on the line ( always residing on the plane). It's essential to understand that any linear combination within a plane remains inside that plane, reinforcing that the combinations have no escape. This principle is very important in showing that every point on the line connecting \(P_1\) and \(P_2\) will also remain in the plane.

Parameterization of a Line

When we talk about the parameterization of a line, we are referring to how we can define it using parameters—in our case, the parameter \(t\). This concept is vital in turning an abstract line into a more tangible form by giving it a direction and magnitude.
Here's how it works:

• You start at an initial point, which we call \(P_1\).
• You then decide a direction to move in, which is established by the vector \(P_2 - P_1\).
• The parameter \(t\) decides how far along in that direction you go.

The key takeaway is that the parameter \(t\) isn't just a random number. It has meaningful control over your location on the line. By tweaking \(t\), you explore different coordinates on that line. This method effectively maps out the line using one variable, showcasing the functionalities of mathematical modeling to describe geometric entities efficiently.