Problem 39 - Calculus for Biology and Medicine

Question

Find the equation of the plane through \((0,0,0)\) and perpendicular to \([1,1,1]\) '.

Step-by-Step Solution

Answer

The equation of the plane is \\(x + y + z = 0\\).

Step 1: Understanding the Problem

The task is to find the equation of a plane given certain conditions. We know the plane passes through the point \(0,0,0\) and is perpendicular to the vector \[1, 1, 1\]. The vector that defines the perpendicular direction is also the normal vector of the plane.

Step 2: Equation of the Plane

The general equation of a plane with a normal vector \(\mathbf{n} = (a, b, c)\) and passing through a point \(P_0(x_0, y_0, z_0)\) is given by \((x-x_0)a + (y-y_0)b + (z-z_0)c = 0\). For this problem, \((x_0, y_0, z_0) = (0,0,0)\), and \((a, b, c) = (1, 1, 1)\).

Step 3: Substituting into the Plane Equation

We substitute the values \(x_0=0\), \(y_0=0\), \(z_0=0\), \(a=1\), \(b=1\), \(c=1\) into the plane equation. This results in \(1(x-0) + 1(y-0) + 1(z-0) = 0\).

Step 4: Simplifying the Equation

Simplifying the equation, we get \(x + y + z = 0\). This equation represents the plane that passes through the origin and is perpendicular to the vector \[1, 1, 1\].

Key Concepts

Understanding the Normal Vector

In three-dimensional geometry, a **normal vector** is a vector that is perpendicular to a given surface, such as a plane. When working with planes, the normal vector is crucial because it defines the plane's orientation in space. For any plane to be uniquely identified by its equation, a normal vector must be provided.

A **normal vector** is generally represented as \((a, b, c)\), where:

• \(a\), \(b\), and \(c\) are the vector components that determine the direction of the perpendicular line to the plane.
• These components also become the coefficients in the equation of the plane.

In the context of our exercise, the normal vector is \((1, 1, 1)\), indicating that the plane is oriented such that it is perpendicular to any vector in the direction of \([1, 1, 1]\). Understanding the direction and components of a normal vector is essential for accurately writing or interpreting the equation of a plane.

Defining the Perpendicular Vector

A plane is defined such that every vector lying on the plane is **perpendicular** to the normal vector. In geometry, perpendicularity implies a 90-degree angle between two vectors. This perpendicular relationship is fundamental to understanding how planes are situated in three-dimensional space.

To determine if a vector is perpendicular to a plane, consider the following:

• The dot product of the vector with the normal vector of the plane should be zero.
• If the dot product is zero, it confirms that the vector is perpendicular to the plane.

For example, in our exercise, the plane is perpendicular to the vector \((1, 1, 1)\). This means any vector, such as \((1, 1, 1)\), that is not on the plane and is in the direction of the perpendicular vector will satisfy this condition, affirming the position of the plane in space.

Exploring Three-Dimensional Coordinates

In three-dimensional geometry, points, lines, and planes are described using **three-dimensional coordinates**. These coordinates, typically given as \((x, y, z)\), represent a point's position in three-dimensional space. Understanding these coordinates is essential when dealing with planes, vectors, and other geometrical entities.

The equation of a plane in three dimensions is of crucial importance because:

• It helps in differentiating a plane from other surfaces.
• It provides a mathematical way to describe all points lying on the plane.

For instance, if you are given the coordinates of a point \((x_0, y_0, z_0)\) and a normal vector \((a, b, c)\), you can find the plane equation by forming the equation \((x-x_0)a + (y-y_0)b + (z-z_0)c = 0\). In our example, these coordinates were \((0, 0, 0)\) and formed the equation \(x + y + z = 0\). Recognizing how three-dimensional coordinates tie together with vectors and planes lets you navigate spatial problems with greater ease and understanding.