Problem 182 - Calculus 3

Question

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right),\) and a vector \(\mathbf{n}=\langle a, b, c\rangle \) that is parallel to the line. Then the equation \(x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t . )\) $$-3 x+9 y+4 z=-4, P(1,-1,2)$$

Step-by-Step Solution

Answer

The parametric equations are \(x = 1 - 3t\), \(y = -1 + 9t\), \(z = 2 + 4t\).

Step 1: Identify the Surface Equation and Point

The given surface equation is \(-3x + 9y + 4z = -4\) and the indicated point on the surface is \(P(1, -1, 2)\).

Step 2: Find the Normal Vector

To find the normal vector to the surface, extract the coefficients of \(x\), \(y\), and \(z\) from the surface equation. The normal vector \(\mathbf{n}\) will be \(\langle -3, 9, 4 \rangle\). This vector is normal to the plane because the given equation is in the form \(ax + by + cz = d\).

Step 3: Write the Parametric Equations for the Normal Line

Using the point \((1, -1, 2)\) and the normal vector \(\langle -3, 9, 4 \rangle\), set up the parametric equations of the line. Use the format: - \(x - x_0 = at\)- \(y - y_0 = bt\)- \(z - z_0 = ct\)Substitute the values:- \(x = 1 - 3t\)- \(y = -1 + 9t\)- \(z = 2 + 4t\)

Step 4: Verify the Parametric Equations

Verify that these parametric equations represent a line passing through the point \((1, -1, 2)\) with direction parallel to \(\langle -3, 9, 4 \rangle\). Substituting \(t = 0\) into the parametric equations should result in the point \((1, -1, 2)\), verifying they are correct.

Key Concepts

Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points making up a geometric object, using a parameter. In the context of lines in space, these equations help us describe a line using one parameter, often denoted by \( t \).
Such equations break down the complex notion of a line into more manageable parts by:

• Defining each coordinate \((x, y, z)\) as a function of the parameter \( t \)
• Illustrating how the line progresses in each coordinate direction

When constructing parametric equations for a normal line, you require a point that the line passes through and a direction, provided by the normal vector. Here, the parametric equations follow the pattern \(x = x_0 + at\), \(y = y_0 + bt\), and \(z = z_0 + ct\), where \((x_0, y_0, z_0)\) is the known point, and \(\langle a, b, c \rangle\) is the normal vector. This simplifies the task of finding the line's equation in space.

Surface Equation

The surface equation represents a flat two-dimensional plane in three-dimensional space. Each plane can be expressed using an equation in three variables \(x\), \(y\), and \(z\), generally in the form \(ax + by + cz = d\).
The surface equation given in the exercise is \(-3x + 9y + 4z = -4\).
This form highlights a plane by establishing:

• The relationship between variables that lies perfectly flat in space
• Coefficients \((-3, 9, 4)\) that indicate the orientation of the normal to the surface of the plane

The constant \(-4\) shifts the plane from including the origin to a specific alignment in space. Understanding these components allows us to explore how the surface interacts with lines and points, and how we can derive additional properties, like the normal vector.

Normal Vector

A normal vector is a key concept in geometry, especially when dealing with surfaces. It is a vector that is perpendicular to a surface or a plane. In this problem, the task is to find the normal vector from the surface equation \(-3x + 9y + 4z = -4\).
The coefficients in the equation, \(-3\), \(9\), and \(4\), directly give us the components of the normal vector, \(\langle -3, 9, 4 \rangle\). This vector:

• Is essential for deriving lines that are perpendicular or normal to the plane
• Serves as a direction for constructing parametric equations of the normal line

By understanding the normal vector, you can gain insights into how the surface orients in its three-dimensional environment and use it to solve problems involving angles and distances of planes.

Points in Space

Points in space are fundamental concepts when navigating three-dimensional geometry. A point is defined by coordinates \((x, y, z)\), where each indicates a location along one of the axes in 3D space.
In our exercise, the point \((1, -1, 2)\) is significant because it lies on the surface defined by the equation.
This point's position influences:

• The particular location where the normal line originates
• How the normal vector and parametric equations are set relative to this specific spot

Without identifying a precise point, it is impossible to establish a unique line. Points provide a spatial reference that helps in constructing geometric figures and in analyzing their relationships, such as intersections with surfaces.