Problem 275 - Calculus Volume 3

Question

For the following exercises, find parametric descriptions for the following surfaces. Plane \(2 x-4 y+3 z=16\)

Step-by-Step Solution

Answer

The plane can be described parametrically as \(x = 10t + \frac{13}{2}s\), \(y = t\), \(z = \frac{16}{3} + s\), where \(t, s\in \mathbb{R}\).

Step 1: Identify the Normal Vector and a Point on the Plane

The equation of the plane is given by \(2x - 4y + 3z = 16\). From this equation, we can extract the normal vector of the plane as \(\langle 2, -4, 3 \rangle\). To find a point on the plane, set two of the variables to zero. Let \(x = 0\) and \(y = 0\): \[ 2(0) - 4(0) + 3z = 16 \] This simplifies to \(3z = 16\) so \(z = \frac{16}{3}\). Thus, a point on the plane is \((0, 0, \frac{16}{3})\).

Step 2: Find Two Direction Vectors

To describe a plane in parametric terms, we need two non-parallel direction vectors on the plane. First, set \(y = 1\), \(z = 0\), and solve for \(x\):\[ 2x - 4(1) + 3(0) = 16 \] which simplifies to \(2x - 4 = 16\) or \(2x = 20\), yielding \(x = 10\). This gives the direction vector \(\mathbf{d_1} = \langle 10, 1, 0 \rangle\). Next, set \(y = 0\), \(z = 1\): \[ 2x - 4(0) + 3(1) = 16 \] which simplifies to \(2x + 3 = 16\) or \(2x = 13\), yielding \(x = \frac{13}{2}\). This gives the direction vector \(\mathbf{d_2} = \langle \frac{13}{2}, 0, 1 \rangle\).

Step 3: Write the Parametric Form of the Plane

With the point \((0, 0, \frac{16}{3})\) and the direction vectors \(\mathbf{d_1} = \langle 10, 1, 0 \rangle\) and \(\mathbf{d_2} = \langle \frac{13}{2}, 0, 1 \rangle\), the parametric equations of the plane are:\[ \begin{align*} x &= 0 + 10t + \frac{13}{2}s \\y &= 0 + 1t + 0s \\z &= \frac{16}{3} + 0t + 1s \end{align*} \] Here, \(t\) and \(s\) are parameters, and the equations describe all points \((x, y, z)\) on the plane.

Key Concepts

Planes in Three-Dimensional Space

In three-dimensional geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions. You can visualize a plane like a giant sheet of paper that has no thickness. It is defined by a formula involving three variables, usually referred to as \(x\), \(y\), and \(z\), which represent coordinates in the 3D space. The general equation of a plane takes the form \(Ax + By + Cz = D \), where \(A\), \(B\), and \(C\) are coefficients that determine the orientation of the plane, and \(D\) is a constant.

This equation is used to express a constraint in space, meaning that any point \((x, y, z)\) that satisfies it will lie on the plane. For the plane \(2x - 4y + 3z = 16\), each combination of \(x\), \(y\), and \(z\) that meets this criterion is a point on the plane.

Planes can also be represented parametrically, using a point on the plane and two direction vectors. This parametric representation allows us to express the plane as a set of equations that describe all the points on the plane in terms of parameters.

Vector Mathematics

Vectors are essential tools in mathematics, especially in physics and engineering.
They are directed quantities that have both magnitude and direction, which means they can represent diverse elements such as force or velocity.

Imagine a vector as an arrow pointing from one point to another in space.

• A vector in three-dimensional space is often denoted by \(\langle x, y, z\rangle\), where \(x\), \(y\), and \(z\) are its components along the respective axes.
• The operations performed on vectors include addition, subtraction, and scalar multiplication.

Vector mathematics is a key component to understanding parametric equations of planes and other geometrical concepts. The parametric form of the plane uses vectors to describe it accurately and fully.

In the original solution, the direction vectors \(\mathbf{d_1} = \langle 10, 1, 0 \rangle\) and \(\mathbf{d_2} = \langle \frac{13}{2}, 0, 1 \rangle\) are examples of how vectors can be used to articulate the nature of a plane. These vectors, alongside a point on the plane, form the complete description of the plane in parametric terms.

Normal Vectors

Normal vectors play a crucial role in understanding the orientation of a plane in space.
A normal vector is a vector that is perpendicular to the plane, acting like a pole standing at a right angle to it.

For any plane given by the equation \(Ax + By + Cz = D\), the vector \(\langle A, B, C\rangle\) serves as the normal vector.

• The normal vector dictates how the plane is positioned in space. Changing the direction of the normal vector rotates the plane around its normal.
• In the example \(2x - 4y + 3z = 16\), the normal vector is \(\langle 2, -4, 3\rangle\). This vector is vital to determine the plane's position and orientation.

Understanding normal vectors is essential for anyone working with planes, as they provide information on how to effectively work and manipulate plane equations. They are often used in calculations involving reflections, projections, and determining angles between planes.