Problem 28 - Algebra and Trigonometry

Question

A description of a line is given. Find parametric equations for the line. The line crosses the \(x\) -axis where \(x=-2\) and crosses the \(z\) -axis where \(z=10\) .

Step-by-Step Solution

Answer

The parametric equations are \( x(t) = -2 + 2t \), \( y(t) = 0 \), and \( z(t) = 10t \).

Step 1: Identify Two Points

The line crosses the x-axis where \( x = -2 \) and the z-axis where \( z = 10 \). This information gives us two points on the line. The point on the x-axis can be written as \((-2, 0, 0)\) since \( y = 0 \) at the x-axis. Similarly, the point on the z-axis is \((0, 0, 10)\) because \( y = 0 \) and \( x = 0 \) on the z-axis.

Step 2: Determine the Direction Vector

To find the direction vector of the line, subtract the coordinates of the point on the x-axis from the coordinates of the point on the z-axis: \( (0, 0, 10) - (-2, 0, 0) = (2, 0, 10) \). Thus, the direction vector of the line \( \vec{d} \) is \( (2, 0, 10) \).

Step 3: Write the Parametric Equations

With the point on the x-axis \((-2, 0, 0)\) as the starting point and \( (2, 0, 10) \) as the direction vector, the parametric equations can be written. Let \( t \) be the parameter. The equations are: \( x(t) = -2 + 2t \), \( y(t) = 0 + 0t = 0 \), and \( z(t) = 0 + 10t \).

Step 4: Verify the Parametric Equations

Substitute \( t = 0 \) into the parametric equations. This gives \( x(0) = -2 \), \( y(0) = 0 \), and \( z(0) = 0 \), which corresponds to the known point on the x-axis. Substituting \( t = 1 \) gives \( x(1) = 0 \), \( y(1) = 0 \), and \( z(1) = 10 \), which matches the known point on the z-axis. Therefore, the calculations are verified.

Key Concepts

Direction Vector

In geometry, a direction vector is crucial for defining a line through space. When a line is described in three dimensions, the direction vector indicates the line's magnitude and orientation. This vector is derived by connecting two distinct points that lie on the line. For example, consider two points, \((-2, 0, 0)\) and \( (0, 0, 10)\), on the same line.
The direction vector is calculated by subtracting the coordinates of the starting point from the coordinates of the ending point. In this case:

• Starting point: \((-2,0,0)\)
• Ending point: \( (0,0,10)\)

The subtraction \( (0,0,10) - (-2,0,0) = (2,0,10)\) provides the direction vector. \(\vec{d} = (2,0,10)\).
This direction vector \(\vec{d}\) serves as a directional "arrow" that projects the line forward or backward across the 3D space, guiding us in constructing the line's parametric equations.

3D Coordinate System

The 3D coordinate system is a framework that enables us to conceptualize geometric figures in three-dimensional space. It involves three axes—typically labeled as \(x\), \(y\), and \(z\)—which are perpendicular to each other. Each point in this system is represented by an ordered triplet \( (x, y, z) \) that locates its position in space.
Here's a closer look at how the axes are structured:

• \(x\)-axis: This axis runs left to right, and all points on it have their \(y\) and \(z\) coordinates as zero.
• \(y\)-axis: Vertically-oriented, this axis has points with zero \(x\) and \(z\) values.
• \(z\)-axis: Extending in depth, points lying only on the \(z\)-axis have \(x\) and \(y\) coordinates of zero.

Within this 3D space, understanding how to navigate using these axes can demystify problems that involve lines and shapes. Any line can be described using the coordinates of points and vectors that align with this system, providing a spatial orientation model for solving complex geometric problems.

Line Intersections

Line intersections in a 3D space refer to the points where lines cross each other or intersect with the coordinate axes. Identifying these intersections requires analyzing the parametric equations of the lines. These intersections provide the exact points where the line meets or crosses another line or plane.
For instance, if a line crosses the \(x\)-axis at \(x = -2\), and the \(z\)-axis at \(z = 10\), it informs us about specific intersection points, namely \((-2, 0, 0)\) and \( (0, 0, 10)\).

• The point \((-2, 0, 0)\) indicates that the line intersects the \(x\)-axis when both \(y\) and \(z\) are zero.
• The point \( (0, 0, 10)\) shows intersection with the \(z\)-axis, where \(x\) and \(y\) are zero.

Using parametric equations helps verify these points by substituting different values in place of the parameter \( t \). Calculations that confirm the line's passing through these intersections on the axes ensure correctness in understanding the line's trajectory and behavior in the 3D plane.