Problem 41 - Calculus for Biology and Medicine

Question

Find the equation of the plane through \((0,0,0)\) and perpendicular to \([1,0,0]^{\prime}\).

Step-by-Step Solution

Answer

The equation of the plane is \\ x = 0 \\.

Step 1: Understanding the Problem

The problem requires us to find the equation of a plane. We are given a point through which the plane passes: \(0,0,0\), and a vector \(1,0,0\) that is perpendicular to the plane.

Step 2: Identifying the Normal Vector

The vector \(1,0,0\) is provided as perpendicular to the plane. In plane equations, this vector serves as the normal vector of the plane. As such, we identify \(1,0,0\) as the normal vector of our plane.

Step 3: Using the Point-Normal Formulation

The general form of a plane through point \((x_0, y_0, z_0)\) and normal \(a, b, c\) is: \ a(x-x_0) + b(y-y_0) + c(z-z_0) = 0 \. Plugging the normal (1,0,0) and point (0,0,0) in, we have \(1(x - 0) + 0(y - 0) + 0(z - 0) = 0\).

Step 4: Solving the Plane Equation

Simplify the equation \(1(x - 0) + 0(y - 0) + 0(z - 0) = 0\) to \ x = 0 \ . This is the equation which represents the plane perpendicular to \(1,0,0\) and passing through the origin.

Key Concepts

Normal Vector

In geometry, particularly in three-dimensional space, the normal vector is crucial for describing planes and surfaces. It's a vector that is perpendicular to the plane in question. Imagine the surface of a table; the normal vector would be like an arrow pointing straight up from the table's surface.

This vector plays a key role when determining the equation of a plane. In our exercise, the vector \(1,0,0\) is the normal vector. It indicates that the plane is perpendicular to the x-axis. Other points on this plane remain constant in terms of their x-coordinate, since the plane essentially "faces" in the direction of this vector.

Here’s a quick list of what the normal vector does:

• Defines the orientation of the plane relative to the coordinate axes.
• Helps in forming the equation of the plane.
• Remains perpendicular to every line lying on the plane.

Understanding the normal vector aids in visualizing how the plane is positioned and why it appears as it does based on its mathematical representation.

Point-Normal Form

The point-normal form is a way to express the equation of a plane using a known point through which the plane passes and its normal vector.

This form is super helpful because it simply ties all these elements together in a formula that expresses the plane's layout in space. Formally, the equation is written as: \[a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\], where \(a, b, c\) are the components of the normal vector, and \(x_0, y_0, z_0\) are the coordinates of the point.

In the exercise, we were given the point \(0,0,0\), and the normal vector \(1,0,0\). Plugging these into the point-normal form equation led us to \(1(x-0) + 0(y-0) + 0(z-0) = 0\), which simplifies further to \(x = 0\).

This representation is simple and gives an immediate understanding of the plane's behavior in space. The point-normal form is fantastic because:

• It uses readily available information (a point and a vector) to define a plane.
• Provides an efficient way to compute the equation of the plane.
• Helps in solving more complex geometric problems by breaking them down into fundamental parts.

Perpendicular Vector

A perpendicular vector, in the context of plane equations, is a vector that stands at a right angle to another vector or surface. In our exercise, the vector \(1,0,0\) is such a vector because it is perpendicular to the plane in question. Technically speaking, when a vector is perpendicular, it forms a 90-degree angle with the surface.

This concept of perpendicularity is the backbone of defining planes. It ensures that any vector lying within the plane will have a dot product of zero with the normal vector. To clarify, here are some insights:

• A vector is perpendicular to a plane if its dot product with every vector in the plane is zero.
• In practical terms, a perpendicular vector helps in navigating and computing distances and intersection in multi-dimensional spaces.
• It's a universal concept in geometry that simplifies analysis of shapes and angles.

Recognizing and understanding perpendicular vectors provides a strong foundation for tackling various geometric and spatial problems. It’s a fundamental idea that opens up more advanced topics in both mathematics and physics.