91Ó°ÊÓ

In mathematics, a plane equation describes a flat, two-dimensional surface that extends indefinitely within a three-dimensional (3D) space. Planes are foundational in geometry and vector calculus. A plane equation is typically written in the general form:\[ ax + by + cz + d = 0 \]Here, \(a, b,\) and \(c\) are coefficients that define the orientation of the plane, and \(d\) determines its specific position in space. In the plane equation for the level surface given in the exercise:- The coefficients are \(a = 3, b = -5,\) and \(c = 3\), - The constant term is \(d = -15\).These coefficients tell us about the plane's tilt relative to the axes. For example, if you were to graph this equation, these numbers direct how the plane slants and rotates as it cuts through the 3D axes.

Three-dimensional space, often simply called 3D space, is the environment within which our world is located and where we observe objects' positions. It is defined by three axes - typically the x, y, and z axes. Every point in 3D space can be represented as a coordinate \((x, y, z)\). This gives a precise location and integrates seamlessly with the concept of a plane.In our exercise's function, the plane we derived \(3x - 5y + 3z - 15 = 0\) exists within this 3D world. The equation represents a flat surface in this three-axes system. So, whenever you visualize how the plane cuts through the 3D space, you're imagining it slicing in a way that perfectly aligns to the calculated intersections with the axes, forming lines and points along the way.

When a plane intersects with one of the axes in 3D space, it typically does so at a distinct point. This gives a tangible location where the plane meets the axis lines. For our specific plane equation \(3x - 5y + 3z - 15 = 0\), it intersects:

• The x-axis when \(y = 0\) and \(z = 0\). Solving, it intersects at the point \((5, 0, 0)\).
• The y-axis when \(x = 0\) and \(z = 0\). Solving, it intersects at the point \((0, 3, 0)\).
• The z-axis when \(x = 0\) and \(y = 0\). Solving, it intersects at the point \((0, 0, 5)\).

These intersections are crucial for understanding both the dimension and location of the plane. They also help in visualizing the extent of the plane's reach within the 3D space, showing how it slices through each axis to define its boundaries.