Problem 2 - Thomas Calculus

Question

In Exercises \(1-12,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$ x=-1, \quad z=0 $$

Step-by-Step Solution

Answer

The set of points is a line parallel to the y-axis at x = -1 and z = 0.

Step 1: Understand the Equations

The given equations are \(x = -1\) and \(z = 0\). These equations describe conditions that the coordinates of points in space must satisfy. \(x = -1\) means the \(x\)-coordinate is constantly \(-1\). Similarly, \(z = 0\) means the \(z\)-coordinate is constantly \(0\).

Step 2: Identify Geometric Constraints

Since \(x = -1\), we are dealing with points that lie on the plane where every \(x\)-value is \(-1\). Since \(z = 0\), we are further constrained to the plane where every \(z\)-value is \(0\).

Step 3: Find Intersection of the Planes

The intersection of the plane \(x = -1\) (a vertical plane parallel to the yz-plane) and the plane \(z = 0\) (a horizontal plane coinciding with the xy-plane) will form a line. Specifically, this line runs along the \(y\)-axis where \(x = -1\) and \(z = 0\).

Step 4: Describe the Geometric Object

The set of all points that satisfies \(x = -1\) and \(z = 0\) is a line that is parallel to the \(y\)-axis. This line is located at an \(x\)-coordinate of \(-1\) and a \(z\)-coordinate of \(0\). The line's equation is \(x = -1\), \(z = 0\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. It merges algebra and geometry, allowing us to describe geometric shapes using equations. In three-dimensional space, every point can be defined by three coordinates:

• \( x \) - the horizontal axis
• \( y \) - the vertical axis
• \( z \) - the depth axis

With coordinate geometry, we can express complex shapes and solve geometric problems by manipulating algebraic equations. For instance, by setting specific coordinate values as in the equations \( x = -1 \) and \( z = 0 \), we can uniquely determine the position of certain points or lines in space. This use of coordinates makes it easier to visualize and calculate the properties of lines, planes, and other shapes in three-dimensional spaces.

Plane Intersection

A plane intersection in geometry refers to the region or line where two planes meet. In three-dimensional space, planes are flat, two-dimensional surfaces extending infinitely. When two planes intersect, they typically form a line.
Consider the planes given by the equations \( x = -1 \) and \( z = 0 \):

• The plane \( x = -1 \) is vertical, parallel to the \( yz \)-plane, and it includes all points where the \( x \)-coordinate is exactly \(-1\).
• The plane \( z = 0 \) is horizontal, coinciding with the \( xy \)-plane, including all points where the \( z \)-coordinate is \(0\).

When these planes intersect, they form a line, specifically where both conditions are satisfied simultaneously. This results in a line where the coordinates are \( x = -1 \) and \( z = 0 \), meaning the line runs parallel to the \( y \)-axis.

Lines in 3D Space

In three-dimensional space, lines are defined by specific conditions on coordinates. A line may be described by one or more equations that express relationships between coordinates.
For example, the line formed by the intersection of planes \( x = -1 \) and \( z = 0 \) follows the conditions:

• \( x = -1 \), making the line parallel to the \( yz \)-plane and fixed at an \( x \)-coordinate of \(-1\).
• \( z = 0 \), placing the line on the \( xy \)-plane.

Thus, this particular line does not deviate along the \( x \)- or \( z \)-axis, but runs infinitely parallel to the \( y \)-axis. Visualizing lines in 3D involves understanding how they interact with the three axes, and using coordinate equations helps us to precisely identify these lines.