Question

Describe with a sketch the sets of points \((x, y, z)\) satisfying the following equations. $$y-z=0$$

Short answer

Answer: A plane in 3D space.

Step-by-step solution

Identify the plane

The given equation is \(y - z = 0\), which can be rewritten as \(y = z\). This equation represents a plane in 3D space where the y-coordinate of any point on the plane is equal to its z-coordinate.

Sketching the plane

Begin by sketching the three axes: \(x\), \(y\), and \(z\). To sketch the plane, find the intersections with each axis. For the intersection with the \(x\)-axis, \(y = z = 0\). This intersection is at the point \((x, 0, 0)\), which occurs at any point along the x-axis. For the intersection with the \(y\)-axis, \(x = z = 0\), so this intersection is at the point \((0, y, 0)\). As \(y=z\), the intersection happens at the point \((0,0,0)\). Finally, for the intersection with the \(z\)-axis, \(x = y = 0\). This intersection is at the point \((0, 0, z)\), but since \(y=z\), this intersection happens at the point \((0,0,0)\). As the plane intersects all three axes at the origin, and the x-axis along every point, our plane is formed by connecting the origin (0,0,0) to any point found on the x-axis and following the condition \(y=z\). By connecting these points, we create a plane that slants upward in a 45-degree angle from the x-axis, with the z-axis as a reflection line. Sketch this plane in your coordinate system. The final sketch will have a plane that passes through every point along the x-axis and where \(y=z\), with a 45-degree angle between the y and z axes.