Problem 16 - Essential Calculus Early Transcendentals

Question

Find a vector equation for the line segment from \((2,-1,4)\) to \((4,6,1) .\)

Step-by-Step Solution

Answer

The vector equation is \( \mathbf{r}(t) = (2 + 2t, -1 + 7t, 4 - 3t) \).

Step 1: Understand the Line Segment

A line segment from point \( A \) to point \( B \) can be represented by a vector equation. Here, \( A = (2, -1, 4) \) is the initial point and \( B = (4, 6, 1) \) is the final point.

Step 2: Determine the Direction Vector

To find the direction vector \( \mathbf{d} \), subtract the coordinates of the initial point \( A \) from the coordinates of the final point \( B \). \( \mathbf{d} = (4 - 2, 6 - (-1), 1 - 4) = (2, 7, -3) \).

Step 3: Write the Vector Equation

The vector equation of a line segment is given by \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{d} \), where \( \mathbf{a} \) is the position vector of the initial point and \( t \) is a parameter that varies from 0 to 1. Substitute the initial point \( \mathbf{a} = (2, -1, 4) \) and direction vector \( \mathbf{d} = (2, 7, -3) \): \[ \mathbf{r}(t) = (2, -1, 4) + t(2, 7, -3) \].

Step 4: Simplify the Vector Equation

Expand the equation \( \mathbf{r}(t) = (2, -1, 4) + t(2, 7, -3) \) to get the vector form: \( \mathbf{r}(t) = (2 + 2t, -1 + 7t, 4 - 3t) \). This represents the entire line segment from the point \( (2, -1, 4) \) to the point \( (4, 6, 1) \) as \( t \) varies from 0 to 1.

Key Concepts

Line Segment

A line segment is a part of a straight line that has two endpoints, connecting them directly. In the world of vectors, this is a powerful concept because it allows us to describe the path between any two points using a simple equation. In our exercise, we have a line segment between two specific points:

• Initial Point: \( A = (2, -1, 4) \)
• Final Point: \( B = (4, 6, 1) \)

A line segment is unique because it has a defined length and it doesn't extend infinitely, as lines typically do. We use the vector equation to describe this line segment because it provides a concise way to express the entire segment mathematically. This is especially useful in geometry and physics, where we often need to define such segments in space.

Direction Vector

To understand how a line segment is represented as an equation, we need to know about the direction vector. This vector specifies the direction in which our line points. It is crucial because it defines the path from the initial point to the final point of the line segment.To find the direction vector \( \mathbf{d} \), we subtract the coordinates of the initial point \( A \) from those of the final point \( B \):

• \( \mathbf{d} = (4 - 2, 6 - (-1), 1 - 4) \)
• \( \mathbf{d} = (2, 7, -3) \)

This direction vector, \( (2, 7, -3) \), tells us how far and in which direction we move from \( A \) to reach \( B \). It's like the GPS directions from point A to point B.

Position Vector

The position vector serves as a starting point in the vector equation of the line segment. It's the reference point from which we begin defining our line in space. In our exercise, the position vector is derived from the initial point of the line segment.The position vector is also known as the initial point's vector, \( \mathbf{a} \), and in our case, it is:

• \( \mathbf{a} = (2, -1, 4) \)

This vector represents the exact location of the initial endpoint of our line segment in a coordinate system. In terms of the vector equation, it anchors the line at this starting point from which we extend our direction vector.By combining the position vector with the direction vector in the equation \( \mathbf{r}(t) = \mathbf{a} + t \mathbf{d} \), we can describe every single point on the line segment as \( t \) changes from 0 to 1, smoothly transitioning from the start point to the endpoint.