Problem 2 - Algebra and Trigonometry

Question

The plane containing the point \(P\left(x_{0}, y_{0}, z_{0}\right)\) and having the normal vector \(\mathbf{n}=\langle a, b, c\rangle\) is described algebraically by the equation ______

Step-by-Step Solution

Answer

The equation is \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\).

Step 1: Find the General Equation of a Plane

The equation of a plane in 3D space with a normal vector \(\mathbf{n}=\langle a, b, c \rangle\) is given by \( ax + by + cz = d \). Here, \(d\) is a constant.

Step 2: Use the Normal Vector and Point to Determine the Plane Equation

To find the constant \(d\), we use the point \(P(x_0, y_0, z_0)\) lying on the plane. Substitute this point into the general equation \( ax + by + cz = d \).

Step 3: Substitute the Point into the Plane Equation

Substitute \(x_0\), \(y_0\), and \(z_0\) into the equation to get \(a(x_0) + b(y_0) + c(z_0) = d \).

Step 4: Write the Specific Plane Equation

After substituting the point, the specific equation of the plane becomes \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\).

Key Concepts

Normal Vector

Understanding the normal vector is key when working with plane equations in 3D space. A normal vector is a vector that is perpendicular to a surface, such as a plane. In the context of a plane, this normal vector is denoted as \( \mathbf{n} = \langle a, b, c \rangle \). This vector essentially defines the orientation or direction of the plane in 3-dimensional space.

When you think of a plane floating in space, this vector sticks out directly from the plane, pointing in a direction perpendicular to it. The components \(a, b,\) and \(c\) of the normal vector play a crucial role in the plane's equation, as they determine how the plane tilts or orients in the space.

By applying the normal vector to the plane's equation, given by \( ax + by + cz = d \), we can see how the vector's components relate directly to the coefficients of \(x, y,\) and \(z\). It's like these components are guiding the plane's position and slope in space.

3D Space

In 3D space, objects are represented not just by their height and width, as in 2D, but also by their depth. This third dimension makes it possible to model real-world objects more realistically. Planes in 3D space are flat surfaces that extend infinitely in all directions within this realm. Each point on a plane can be defined by three coordinates: \(x\), \(y\), and \(z\).

Understanding 3D space helps in visualizing how planes, lines, and points coexist and interact. For example, consider how a paper (plane) might sit on a desk (other planes). Imagine how a normal vector would point out from the paper, remaining perpendicular no matter how you tilt it.

When you work with equations in 3D, like the equation of a plane \( ax + by + cz = d \), you're essentially describing the relationship between all possible points \((x, y, z)\) on this surface. The equation represents a specific flat surface within the 3-dimensional expanse, characterized by its unique position and orientation.

Constant in the Plane Equation

The constant \(d\) in the equation of a plane \( ax + by + cz = d \) plays a vital role in positioning the plane within 3D space. While the normal vector sets the orientation of the plane, the constant \(d\) determines how far this plane shifts from the origin (0,0,0) of the space.

To find this constant, you can take a point \(P(x_0, y_0, z_0)\) that lies on the plane and substitute its coordinates into the equation, resulting in \( a(x_0) + b(y_0) + c(z_0) = d \). This step ensures that the plane passes through the chosen point, locking its position in space.

This constant can be thought of as the plane's unique fingerprint, ensuring that it does not merely float arbitrarily through space but is anchored exactly where it should be. Essentially, \(d\) helps define the precise spatial location of the plane, complementing the alignment provided by the normal vector.