Problem 46 - Contemporary Precalculus

Question

Use Exercise 44 to find a parameterization of the line segment joining the two points. Confirm your answer by graphing. $$(14,-5) \text { and }(5,-14)$$

Step-by-Step Solution

Answer

Question: Determine the parameterization of the line segment joining the points (14, -5) and (5, -14), and confirm the answer by graphing the line segment. Answer: The parameterization of the line segment joining the points (14, -5) and (5, -14) is given by the formula 𝐫(t) = (14 - 9t, -5 - 9t), for t ∈ [0, 1]. Upon graphing the line segment, it passes through the given points, confirming the correctness of the parameterization.

Step 1: Determine the parameterization formula

The formula for a parameterization of a line segment is: \({\bf r}(t) = (1-t){\bf a} + t{\bf b}\), where \({\bf a}\) and \({\bf b}\) are the endpoints of the line segment, and \(t \in [0, 1]\).

Step 2: Plug in the given points

The given points are \((14, -5)\) and \((5, -14)\). Plugging the coordinates of these points into the parameterization formula, we get: \({\bf r}(t) = (1-t)(14, -5) + t(5, -14)\)

Step 3: Distribute \((1-t)\) and \(t\) to the coordinates

Now we distribute \((1-t)\) and \(t\) to the coordinates of the points: \({\bf r}(t) = ((1-t)14, (1-t)(-5)) + (t5, t(-14))\)

Step 4: Combine the coordinates

Combine the coordinates for the parameterization: \({\bf r}(t) = ((1-t)14 + t5, (1-t)(-5) + t(-14))\)

Step 5: Simplify the expression

Simplify the expression: \({\bf r}(t) = (14 - 14t + 5t, -5 + 5t - 14t)\) \({\bf r}(t) = (14 - 9t, -5 - 9t)\)

Step 6: Confirm the answer by graphing

Now we confirm our answer by graphing the line segment. The graph should show a straight line from \((14, -5)\) to \((5, -14)\). As you can see from the graph, the line segment does indeed pass through the two points. Therefore, our parameterization of the line segment is correct.

Key Concepts

Line Segment Parameterization

Line segment parameterization is a method used to create a simple way to describe a line segment between two points in space using an equation. Imagine having two points, say \((14, -5)\) and \((5, -14)\), and you want to find all the points on the line segment connecting them. This can be done by varying a parameter, typically denoted as \(t\), which smoothly transitions between the two points.
The basic parameterization formula is:

• \(\textbf{r}(t) = (1-t)\textbf{a} + t\textbf{b}\)

where \(\textbf{a}\) and \(\textbf{b}\) are the endpoints of the line segment, and \(t\) is a parameter ranging from 0 to 1. When \(t = 0\), \(\textbf{r}(t)\) is at \(\textbf{a}\), and when \(t = 1\), \(\textbf{r}(t)\) is at \(\textbf{b}\). For values of \(t\) between 0 and 1, \(\textbf{r}(t)\) traces the entire line segment. This method is widely used in computer graphics and physics because it allows straightforward interpolation between two points.

Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities known as vectors. Vectors have both magnitude and direction, in contrast to scalars that only have magnitude. In the context of line segments, vectors make it easier to calculate transitions between points. A vector can be represented as \(\textbf{a} = (a_1, a_2)\) in two-dimensional space.
Using vector operations such as addition and scalar multiplication, we can describe movements along a line or curve. In the parameterization of line segments, vector algebra helps in distributing scalar values like \((1-t)\) and \(t\) over the vector endpoints \(\textbf{a}\) and \(\textbf{b}\).
The significance of vector algebra in line segment parameterization lies in its ability to provide precise calculations for position and direction. Calculating a parameterized line segment as

• \((1-t)\textbf{a} + t\textbf{b}\)

involves adding and scaling vectors, which are fundamental operations in vector algebra. This computation allows us to extract any point on the line segment between two given points.

Two-Point Form

The two-point form is a way to express equations of lines or line segments when you know two points on the line. It is particularly useful when deriving the equation of a line segment between these points, \((x_1, y_1)\) and \((x_2, y_2)\). In this form, you can express the slope, or rate of change, between the points and develop an equation of the line.
For a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \), the two-point form is given by:

• \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \)

This equation explicitly uses the differences in the \(x\) and \(y\) values between the two points to calculate the line's slope. From there, you can find any point on the line by selecting any \(x\) value in the desired range and solving for \(y\).
The two-point form is fundamentally intertwined with line segment parameterization. It ensures that we understand how the points are connected and how the line's slope informs the interpolation between them. While parameterization involves vectors, the two-point form offers another perspective, focusing on slope and individual points.