/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Find an equation of the plane th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 14

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(1,2,-3) ; \mathbf{n}=\langle-1,4,-3\rangle$$

Short Answer

Expert verified
Question: Find the equation of a plane that passes through point \(P_{0}(1,2,-3)\) and has a normal vector \(\mathbf{n} = \langle-1, 4, -3\rangle\). Answer: The equation of the plane is given by \(-x + 4y - 3z = 16\).

Step by step solution

01

Convert Point \(P_{0}\) and Normal Vector \(\mathbf{n}\) to Position Vectors

We are given the point \(P_{0} (1, 2, -3)\) and the normal vector \(\mathbf{n} = \langle-1, 4, -3\rangle\). We can convert these into position vectors by writing each as a column matrix: $$\mathbf{p_0} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \quad \mathbf{n} = \begin{pmatrix} -1 \\ 4 \\ -3 \end{pmatrix}$$
02

Write the Formula for the Equation of a Plane

Recall the general formula for the equation of a plane, given a point and a normal vector: $$(\mathbf{n} \cdot \mathbf{r}) = (\mathbf{n} \cdot \mathbf{r_0})$$ Here, \(\mathbf{r}\) is a vector representing any point on the plane, and we already have position vectors for \(\mathbf{r_0}\) and the normal vector \(\mathbf{n}\).
03

Compute the Dot Product \((\mathbf{n} \cdot \mathbf{r_0})\)

Now, find the dot product of the normal vector \(\mathbf{n}\) and the position vector \(\mathbf{r_0}\): $$(\mathbf{n} \cdot \mathbf{r_0}) = \begin{pmatrix} -1 \\ 4 \\ -3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix} = (-1)(1) + (4)(2) + (-3)(-3) = -1 + 8 + 9 = 16$$
04

Compute the Dot Product \((\mathbf{n} \cdot \mathbf{r})\) and Find the Equation of the Plane

Now, compute the dot product of the normal vector \(\mathbf{n}\) and the vector \(\mathbf{r}\), which represents any point (x, y, z) on the plane: $$(\mathbf{n} \cdot \mathbf{r}) = \begin{pmatrix} -1 \\ 4 \\ -3 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = -x + 4y - 3z$$ Since \((\mathbf{n} \cdot \mathbf{r}) = (\mathbf{n} \cdot \mathbf{r_0})\), we can equate the two expressions we found: $$-x + 4y - 3z = 16$$ This is the final equation of the plane passing through the point \(P_{0}(1, 2, -3)\) and having a normal vector \(\mathbf{n} = \langle-1, 4, -3\rangle\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Vector
In the realm of geometry, particularly in three-dimensional space, the concept of a normal vector is crucial for understanding planes. A normal vector is a vector that is perpendicular to a given surface or plane.
This means it "sticks out" at a right angle from the surface. For a plane in 3D space, the normal vector provides essential information about the plane鈥檚 orientation.
  • The normal vector is often denoted as \(\mathbf{n}\) and is usually expressed in component form, such as \(\langle a, b, c \rangle\).
  • In the equation of a plane \(-x + 4y - 3z = 16\), the coefficients of \(x\), \(y\), and \(z\) directly come from the components of the normal vector \(\mathbf{n} = \langle -1, 4, -3 \rangle\).
  • This vector provides the "directional" information of the plane, indicating how the plane is oriented in the three-dimensional space.
To determine any plane, knowing just a normal vector and a point on the plane is sufficient. This is because the normal vector helps to establish the plane's direction by ensuring it's always perpendicular to any vector lying on the plane.
Dot Product
The dot product, also known as the scalar product, is a fundamental operation in vector mathematics. When you compute the dot product of two vectors, you get a scalar (a single number) rather than another vector. This operation plays a vital role in finding the equation of a plane.
  • The dot product is calculated by multiplying corresponding components of the two vectors and then summing the results.
  • For example, to find \(\mathbf{n} \cdot \mathbf{r_0}\), where \(\mathbf{n} = \langle -1, 4, -3 \rangle\) and \(\mathbf{r_0} = \langle 1, 2, -3 \rangle\), we do the following computation: \((-1)(1) + (4)(2) + (-3)(-3) = 16\).
In the context of determining the equation of a plane, the dot product \(\mathbf{n} \cdot \mathbf{r}\) yields an expression that represents all points \((x, y, z)\) on the plane. By setting this dot product equal to the computed value of \(\mathbf{n} \cdot \mathbf{r_0}\), we derive the plane's equation. This process ensures the plane's equation rightly aligns with its geometric properties.
Position Vectors
Position vectors are an essential concept in vector algebra, often used to locate points in space relative to a fixed origin. These vectors provide a clear way to represent any point in a three-dimensional environment.
  • A position vector extends from the origin \((0,0,0)\) to the specific point.
  • For instance, the position vector for point \(P_0\) (1, 2, -3) is represented as \(\begin{pmatrix} 1 \ 2 \ -3 \end{pmatrix}\).
  • Position vectors help express the location of a point in terms of direction and magnitude.
In the analysis of a plane, position vectors provide a means to specify the reference point \(\mathbf{r_0}\) on the plane. Using the position vector simplifies calculating the dot product with the normal vector, as it organizes the point's coordinates explicitly, which clarifies the relationship within the plane鈥檚 mathematical framework.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(Adapted from 1938 Putnam Exam) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane \(P\) given by \(A x+B y+C z+1=0\). Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}\) and \(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2}\) a. Find the equation of the plane tangent to the ellipsoid at the point \((p, q, r)\) b. Find the two points on the ellipsoid at which the tangent plane is parallel to \(P\) and find equations of the tangent planes. c. Show that the distance between the origin and the plane \(P\) is \(h\) d. Show that the distance between the origin and the tangent planes is \(h m\) e. Find a condition that guarantees that the plane \(P\) does not intersect the ellipsoid.

Find the points (if they exist) at which the following planes and curves intersect. $$\begin{aligned} &2 x+3 y-12 z=0 ; \quad \mathbf{r}(t)=\langle 4 \cos t, 4 \sin t, \cos t\rangle\\\ &\text { for } 0 \leq t \leq 2 \pi \end{aligned}$$

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{-x} \sin y$$

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Generalize the procedure in Exercise 70 by assuming that \(n\) data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) are given. Write the function \(E(m, b)\) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are $$ \begin{aligned} m &=\frac{\left(\sum x_{k}\right)\left(\sum y_{k}\right)-n \sum x_{k} y_{k}}{\left(\sum x_{k}\right)^{2}-n \sum x_{k}^{2}} \text { and } \\ b &=\frac{1}{n}\left(\sum y_{k}-m \Sigma x_{k}\right) \end{aligned}, $$ where all sums run from \(k=1\) to \(k=n\).

Recall that Cartesian and polar coordinates are related through the transformation equations $$\left\\{\begin{array}{l} x=r \cos \theta \\ y=r \sin \theta \end{array} \quad \text { or } \quad\left\\{\begin{array}{l} r^{2}=x^{2}+y^{2} \\ \tan \theta=y / x \end{array}\right.\right.$$ a. Evaluate the partial derivatives \(x_{r}, y_{r}, x_{\theta},\) and \(y_{\theta}\) b. Evaluate the partial derivatives \(r_{x}, r_{y}, \theta_{x},\) and \(\theta_{y}\) c. For a function \(z=f(x, y),\) find \(z_{r}\) and \(z_{\theta},\) where \(x\) and \(y\) are expressed in terms of \(r\) and \(\theta\) d. For a function \(z=g(r, \theta),\) find \(z_{x}\) and \(z_{y},\) where \(r\) and \(\theta\) are expressed in terms of \(x\) and \(y\) e. Show that \(\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.