Problem 5 - Calculus: Early Transcendentals

Question

The equation \(x+y+z=1\) describes some collection of points in \(\mathbb{R}^{3} .\) Describe and sketch the points that satisfy \(x+y+z=1\) and are in the xy- plane, in the xz-plane, and in the yz-plane.

Step-by-Step Solution

Answer

The solutions are lines: \(x+y=1\) in the xy-plane, \(x+z=1\) in the xz-plane, and \(y+z=1\) in the yz-plane.

Step 1: Determine points in the xy-plane

Points in the xy-plane have a z-coordinate of 0. Substitute \(z = 0\) into the equation \(x + y + z = 1\). This simplifies to \(x + y = 1\). This is the equation of a line in the xy-plane.

Step 2: Determine points in the xz-plane

Points in the xz-plane have a y-coordinate of 0. Substitute \(y = 0\) into the equation \(x + y + z = 1\). This simplifies to \(x + z = 1\). This is the equation of a line in the xz-plane.

Step 3: Determine points in the yz-plane

Points in the yz-plane have an x-coordinate of 0. Substitute \(x = 0\) into the equation \(x + y + z = 1\). This simplifies to \(y + z = 1\). This is the equation of a line in the yz-plane.

Step 4: Sketch the lines in their respective planes

Draw a coordinate system for each plane: xy, xz, and yz. Plot the lines \(x + y = 1\) in the xy-plane, \(x + z = 1\) in the xz-plane, and \(y + z = 1\) in the yz-plane. Each line can be represented by two points: for instance, in the xy-plane, plot points (1,0) and (0,1) to draw the line \(x + y = 1\). Follow a similar process for the other two planes.

Key Concepts

three-dimensional geometry

Three-dimensional geometry involves objects that have length, width, and height. Unlike two-dimensional geometry (which deals with shapes like squares or circles on a flat surface), three-dimensional geometry considers the full volume of objects. In the context of coordinate planes, three-dimensional space \(\mathbb{R}^3\) allows us to locate points using three coordinates, typically labeled as \(x, y,\) and \(z\).
In this space, we can describe various geometric shapes and figures through equations that relate \(x\), \(y\), and \(z\).
For example, the equation \(x+y+z=1\) defines a plane, a flat two-dimensional surface, within this three-dimensional space. This plane contains infinite points that satisfy the equation, spreading out in three directions.
In three-dimensional geometry, understanding planes, lines, and points is crucial when exploring the spatial relationships between different figures.

xy-plane

The \(xy\)-plane is an essential concept in three-dimensional geometry.
It's a flat surface where the \(x\) and \(y\) coordinates can vary freely, but the \(z\) coordinate is always zero. You can think of the \(xy\)-plane as the floor of a room, where you can move forward, backward, and sideways, but not up or down.
In the context of the exercise, when we consider the equation \(x + y + z = 1\) within the \(xy\)-plane, we set \(z = 0\). This simplifies the equation to \(x + y = 1\), describing a line within this plane.
On a graph, you can sketch this line by selecting points like \( (1,0)\) and \( (0,1)\) and drawing a straight line through them. This line is the segment of the plane where the equation's conditions are met.

xz-plane

The \(xz\)-plane is another critical plane that you come across in three-dimensional geometry.
This plane allows the \(x\) and \(z\) coordinates to vary freely, with the \(y\) coordinate fixed at zero. Imagine this as a vertical wall where you can climb up and move side to side but cannot move in or out from the wall.
For our equation \(x + y + z = 1\), setting \(y = 0\) means the equation becomes \(x + z = 1\). This equation describes a line in the \(xz\)-plane.
To draw this line, identify points such as \( (1,0)\) and \( (0,1)\) on a graph and connect them with a straight line. This visual representation highlights the points that meet the conditions of the equation within this plane.

yz-plane

The \(yz\)-plane is the third main coordinate plane encountered.
This plane holds the \(y\) and \(z\) coordinates variable while keeping the \(x\) coordinate fixed at zero. Picture this as a side wall you can approach from either up-and-down or sideways directions, but you cannot move forward or backward.
In our equation \(x + y + z = 1\), when \(x = 0\), it simplifies to \(y + z = 1\). This line, when graphed, lies within the \(yz\)-plane.
To visualize it, plot points such as \( (0,1)\) and \( (1,0)\) and join them with a line. This illustrates the line where the equation holds true in this specific plane.