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Understanding the 3D coordinate system is fundamental for visualizing objects in space, such as planes and line segments. Similar to the 2D system with X and Y axes, the 3D coordinate system introduces a third axis—Z—to represent depth. Picture this system as a corner of a room where two walls meet the floor. The point where they all intersect is called the origin, labeled as (0,0,0). From the origin, we have three axes:

• X-axis: It extends horizontally to the right.
• Y-axis: It extends vertically upwards.
• Z-axis: It projects forwards or backwards, coming out of the page towards you or going into the page away from you.

Coordinates in this system are written as triples, for example, (x,y,z), indicating a point's position along the X, Y, and Z axes, respectively. When plotting points, you'll typically start at the origin, move along the X-axis for the x-coordinates, shift parallel to the Y-axis for the y-coordinates, and finally adjust along the Z-axis for the z-coordinates. Multiple points can then define shapes like lines, curves, and planes within this three-dimensional space.

Cartesian equations allow us to represent geometric figures algebraically. For planes in a 3D coordinate system, a standard form of the Cartesian equation is used: Ax + By + Cz + D = 0, where A, B, C, and D are constants. This equation is a rule that every point (x, y, z) on the plane must satisfy.

In the context of the exercise, the given equation may initially seem complex, but it can be transformed into a more accessible form. The equation \[ 3x + 6y - z = 6 \]is manipulated to isolate z on one side, resulting in \[ z = 3x + 6y - 6 \].This form reveals how z is affected by values of x and y. The coefficients 3, 6, and -1 (implied for z) are the direction ratios of the plane, providing a direct measure of the slope or gradient along each of the three axes. By translating the plane's equation into this format, we grasp how changes in the x and y coordinates influence the z-coordinate, which is invaluable for sketching the plane.

Sketching planes involves multiple steps that help us move from a conceptual understanding to a visual representation. We start by identifying specific points that satisfy the plane's equation and using those to anchor our sketch.Taking the rearranged Cartesian equation \[ z = 3x + 6y - 6 \],points on the plane are found by strategically setting two coordinates to zero and solving for the third. This technique yields intercepts with the axes, creating tangible markers for us to plot. For example, setting x and y to 0 gives us the z-intercept (0,0,-6). Similar methods provide the x-intercept (2,0,0) and y-intercept (0,1,0).

Next, we plot these points in our 3D coordinate system—the visual representation of our room corner analogy. Connecting the dots effectively, we sketch lines between the points to form a triangular shape, a portion of the plane. This triangle is part of the plane, and ideally, we should imagine the surface extending indefinitely in all directions. While we can't draw this infinite expanse, we can indicate its presence by shading the triangular region we plotted. This shaded area symbolizes the plane's spread beyond the limits of our paper or screen.