Problem 3 - 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. $$ y=0, \quad z=0 $$

Step-by-Step Solution

Answer

The solution describes the set of points on the x-axis in 3D space.

Step 1: Analyze the first equation

The first equation given is \(y = 0\). This equation represents the set of all points in three-dimensional space where the \(y\)-coordinate is always 0.

Step 2: Analyze the second equation

The second equation is \(z = 0\), meaning this represents the set of all points where the \(z\)-coordinate is always 0 in three-dimensional space.

Step 3: Combine the equations

When both \(y = 0\) and \(z = 0\) are true, both the \(y\)-coordinate and the \(z\)-coordinate of each point are fixed at 0. This means we are considering only points where these two conditions hold.

Step 4: Interpret the geometric description

Points that satisfy the equations \(y = 0\) and \(z = 0\) lie on the \(x\)-axis in three-dimensional space, as the \(x\)-axis consists solely of points where both \(y\) and \(z\) are 0.

Key Concepts

Equations in 3D Space

Understanding equations in three-dimensional space offers an enriched perspective compared to two-dimensional planes. In 3D space, equations can define surfaces or lines in a way that represents the spatial relationships between the three axes: the x-axis, y-axis, and z-axis.

When interpreting equations like \( y = 0 \) and \( z = 0 \), we visualize restrictions on the spatial orientation of points. Each of these equations constrains one of the spatial dimensions to a fixed value, effectively reducing the degrees of freedom for any point satisfying them.

For instance, if only \( y = 0 \) were given, the solution set would be a plane that extends infinitely in the xz-coordinate system at \( y = 0 \). Similarly, \( z = 0 \) alone describes a plane extending in the xy-coordinate system.

Combining equations further confines the solution to lines or intersections within the 3D space, such as when \( y = 0 \) and \( z = 0 \) intersect, creating the x-axis. Equations like these are powerful tools for geometrically describing and analyzing spatial constraints.

Coordinates

Coordinates in 3D space are essential for pinpointing the exact location of a point. Each point in three-dimensional space is identified by a triplet of values: \((x, y, z)\). These values correspond to the point's perpendicular distances from the respective planes YZ, XZ, and XY.

In the context of equations \( y = 0 \) and \( z = 0 \), the coordinates reveal a lot about spatial positioning. Here, all points have their \( y \) and \( z \) values fixed at zero, meaning any variation in position is entirely along the x-axis.

Points are expressed simply as \((x, 0, 0)\), where \( x \) can be any real number.

• The x-coordinate can range from \( -\infty \) to \( +\infty \), covering every potential position on the x-axis.
• The inability to vary in \( y \) or \( z \) direction signifies restriction to a one-dimensional space within the three-dimensional framework.

Understanding coordinates and how they represent points enhances your ability to interpret and visualize spatial relationships in equations comprehensively.

X-axis

The x-axis is a fundamental component of any Cartesian system, acting as one of the primary axes in three-dimensional geometry. It's the reference line along which all points have both their \( y \) and \( z \) coordinates equaling zero.

Interestingly, the x-axis serves as the primary line of reference for movement restricted by the equations \( y = 0 \) and \( z = 0 \). When both conditions are met, all points align with the x-axis, signifying total restriction in both vertical and lateral movement.

The x-axis extends infinitely in both directions along the positive and negative sides of the x value. This property makes it a crucial component in plotting behaviors or transformations of points resulting from these particular sets of equations.

• Movement along the x-axis is solely dictated by the x-coordinate, free from influences that affect y or z.
• Its importance is not just limited to pure geometry but trickles into analytic geometry and vector calculus, where it acts as a key reference line.

Recognizing where the x-axis stands in the grand scheme of 3D equations helps in understanding how such conditions constrain more complex geometric structures.