91Ó°ÊÓ

A plane is a flat, two-dimensional surface that extends infinitely in three-dimensional space. In 3D coordinate geometry, the equation of a plane is often expressed in the form:
\[ ax + by + cz = d \]
where \(a\), \(b\), and \(c\) are the coefficients that determine the orientation of the plane, and \(d\) is a constant. These coefficients also form the components of the normal vector, a key element in plane equations, perpendicular to the plane itself.
For the given points \(A(3, 2, -4)\), \(B(1, 1, -4)\), and \(C(0, 1, -4)\), the calculation in the step-by-step solution shows that the normal vector is \((0, 0, -1)\), leading to the plane equation \(z = -4\). This equation illustrates that the plane is parallel to the \(xy\)-plane and located at \(z = -4\).

Vectors in 3D are directed line segments defined by both a direction and a magnitude. Three-dimensional vectors can be expressed as \((x, y, z)\). These vectors are pivotal in defining direction in space and calculating various geometric properties, such as planes and angles between lines.
For example, in our exercise, the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are calculated to help identify the plane through points \(A, B,\) and \(C\). They were computed as \((−2, −1, 0)\) and \((−3, −1, 0)\) respectively. These vectors lie on the same plane and provide the means to calculate a normal vector via the cross product method. Understanding vectors is crucial, as they help in performing essential 3D geometric operations.

The cross product is a mathematical operation used to find a vector perpendicular to two given vectors in three-dimensional space. If you have two vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\), their cross product \(\overrightarrow{u} \times \overrightarrow{v}\) is calculated using the determinant of a matrix formed by these vectors and the unit vectors \(\hat{i}, \hat{j},\) and \(\hat{k}\).
For our problem, the cross product of vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) results in the normal vector \(\overrightarrow{n} = (0, 0, -1)\). This indicates that \(\overrightarrow{n}\) points in the negative \(z\)-direction, thus defining the orientation of the plane with respect to the coordinate axes. The precision of this operation makes it possible to directly employ the resulting vector in deriving the plane's equation.

A coordinates system in three dimensions is defined by three axes: \(x\), \(y\), and \(z\). Each point in this system is represented by an ordered triplet \((x, y, z)\). This system provides a framework for positioning objects in space and is fundamental in solving problems in 3D geometry.
In the presented exercise, the points \(A(3, 2, -4)\), \(B(1, 1, -4)\), and \(C(0, 1, -4)\) share the same \(z\)-coordinate, showing they lie on a plane parallel to the \(xy\)-plane. Understanding the positioning of points in this 3D system simplifies the process of deriving relationships like plane equations and computing vector operations, thus solidifying the foundation of coordinate geometry for problem-solving in multiple dimensions.