Problem 20 - University Calculus: Early Transcendentals

Question

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$(1,0,-1), \quad(0,3,0)$$

Step-by-Step Solution

Answer

Parametrization: \( (x,y,z) = (1-t, 3t,-1+t) \) for \(0 \leq t \leq 1\).

Step 1: Understand the Problem

We need to find a parametrization for the line segment joining the points \((1,0,-1)\) and \((0,3,0)\). A parametrization represents the line segment as a function of a parameter \(t\).

Step 2: Determine Vector from Points

Calculate the direction vector \(\mathbf{v}\) by subtracting the start point from the end point: \(\mathbf{v} = (0 - 1, 3 - 0, 0 - (-1)) = (-1, 3, 1)\).

Step 3: Choose a Parametrization

Using the vector \(\mathbf{v} = (-1, 3, 1)\) and the starting point \((1, 0, -1)\), write the line segment in parametric form. This is \((1, 0, -1) + t(-1, 3, 1)\) for \(0 \leq t \leq 1\).

Step 4: Write Parametric Equations

From the parametrization, express each coordinate: \(x = 1 - t\), \(y = 3t\), and \(z = -1 + t\). These equations give the set of coordinates for the line segment as \(t\) varies.

Step 5: Interpret the Parametrization

\(t=0\) corresponds to the point \((1, 0, -1)\), and \(t=1\) corresponds to \((0, 3, 0)\). As \(t\) increases from 0 to 1, the line segment moves from the start point to the end point, indicating the direction of increasing \(t\).

Step 6: Sketch the Line Segment

On a 3D coordinate system, plot the points \((1, 0, -1)\) and \((0, 3, 0)\). Draw the directed line segment connecting them. The parameter \(t\) represents a movement from the first point to the second, showing the direction of the line segment.

Key Concepts

Vector Subtraction

Vector subtraction is the process of finding the vector that connects two points in a space. It's essentially about finding a direction and a distance from one point to another. When we have two points, say \(A=(1,0,-1)\) and \(B=(0,3,0)\), the direction from \(A\) to \(B\) can be found using vector subtraction:

• Subtract each coordinate of \(A\) from \(B\): \( (0-1, 3-0, 0-(-1)) \).
• This gives a direction vector \( \mathbf{v} = (-1, 3, 1) \).

This vector tells us how to move from point \(A\) to point \(B\). The components of the vector \(v\) show changes in the x, y, and z directions respectively.

3D Coordinate System

The 3D coordinate system is a way to locate points in space with three numbers or coordinates. Each point is written as \( (x, y, z) \). Here’s a simple breakdown:

• The \(x\)-axis runs left to right,
• The \(y\)-axis runs up and down, and
• The \(z\)-axis runs forward and backward.

This system allows us to plot and sketch lines or vectors. For example, if you have points \( (1,0,-1) \) and \( (0,3,0) \), you place them on these axes. This lets you see how the points and lines relate to each other in three-dimensional space.

Line Segment

A line segment connects two points and includes all the points between these two endpoints. In the context of vector math, a line segment is often described with an equation. We find the equation by starting at one point and following the direction of the vector that was calculated through vector subtraction.In our exercise, the line segment starts at \( (1,0,-1) \) and ends at \( (0,3,0) \). Using the direction vector \( \mathbf{v} = (-1, 3, 1) \), every position on the line segment can be calculated and represented as a combination of these points and movements along the direction vector.

Parametric Equations

Parametric equations are a set of equations that express the coordinates of points on a curve or line in terms of one parameter, usually \(t\). These equations let us navigate along a line segment smoothly. Each coordinate axis (x, y, z) gets its equation based on the parametrization.In our specific case, the equations are:

• \(x = 1 - t\)
• \(y = 3t\)
• \(z = -1 + t\)

As \( t \) varies from 0 to 1, we move along the line segment from \( (1,0,-1) \) to \( (0,3,0) \). When \( t = 0 \), we are at the start point, and when \( t = 1 \), we arrive at the end point. This approach is very useful in computer graphics and physics where modeling of lines and curves is necessary.