/*! 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 27 Determine whether the following ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 27

Determine whether the following pairs of planes are parallel, orthogonal, or neither. $$2 x+2 y-3 z=10 \text { and }-10 x-10 y+15 z=10$$

Short Answer

Expert verified
Question: Determine whether the given pair of planes are parallel, orthogonal, or neither: $$2x + 2y - 3z = 10$$ and $$-10x - 10y + 15z = 10$$. Answer: The given pair of planes are neither parallel nor orthogonal.

Step by step solution

01

Identify Normal Vectors

For the given planes, we can identify their normal vectors by looking at the coefficients of $$x, y, z$$. For the first plane, $$2x + 2y - 3z = 10$$, the normal vector is $$\boldsymbol{n}_1 = (2, 2, -3)$$. For the second plane, $$-10x - 10y + 15z = 10$$, the normal vector is $$\boldsymbol{n}_2 = (-10, -10, 15)$$.
02

Check if the Normal Vectors are Proportional

To see if the normal vectors are proportional, we check if their components satisfy a scalar multiplication relationship. If there exists a scalar $$k$$ such that $$\boldsymbol{n}_1 = k\boldsymbol{n}_2$$, then the normal vectors are proportional, and the planes are parallel. In our case, we have: $$2 = -10k$$ $$2 = -10k$$ $$-3 = 15k$$ From the first equation, we find \(k = -\frac{1}{5}\), but substituting this value into the third equation, we get $$-3\neq 15(-\frac{1}{5}) = -3$$, and thus, the normal vectors are not proportional, and the planes are not parallel.
03

Compute the Dot Product of the Normal Vectors

To compute the dot product of the normal vectors, we just need to multiply their corresponding components and add the results together. $$\boldsymbol{n}_1 \cdot \boldsymbol{n}_2 = (2)(-10) + (2)(-10) + (-3)(15)$$ $$\boldsymbol{n}_1 \cdot \boldsymbol{n}_2 = -20 - 20 + 45 = 5$$ Since the dot product is not equal to zero, the normal vectors are not orthogonal, and thus the planes are not orthogonal.
04

Conclusion

Since the normal vectors are neither proportional nor orthogonal, the given pair of planes are neither parallel nor orthogonal.

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 Vectors
Normal vectors are essential in understanding the orientation of planes in three-dimensional space. In essence, a normal vector is a vector that is perpendicular to a surface or a plane. These vectors help define the plane's orientation in 3D space.
To find a plane's normal vector, look at the coefficients of the variables in a plane equation of the form:
\[ ax + by + cz = d \]
  • The normal vector \( \mathbf{n} = (a, b, c) \) is derived from these coefficients.
  • For the plane equation \( 2x + 2y - 3z = 10 \), the normal vector is \( \mathbf{n}_1 = (2, 2, -3) \).
  • In another plane equation like \( -10x - 10y + 15z = 10 \), the normal vector is \( \mathbf{n}_2 = (-10, -10, 15) \).
Understanding which normal vectors belong to which plane helps when deciding if two planes are parallel or orthogonal.
Parallel Planes
Parallel planes never meet, regardless of how far they extend. For two planes to be parallel, their normal vectors must be proportional. This means that one normal vector can be expressed as a scalar multiple of the other.
To determine if two planes are parallel:
  • Check if there exists a scalar \( k \) such that \( \mathbf{n}_1 = k \mathbf{n}_2 \).
  • If they satisfy this condition, the planes are parallel.
For instance, in our original exercise, check if:
  • \( 2 = -10k \)
  • \( 2 = -10k \)
  • \( -3 = 15k \)
In this case, solving these equations failed to yield a consistent \( k \), meaning the planes aren't parallel. This showcases the core idea: without a valid \( k \), planes will diverge at some point.
Orthogonal Planes
Orthogonal planes intersect at right angles. The key to identifying orthogonal planes is their normal vectors. For two planes to be orthogonal, the dot product of their normal vectors must equal zero.
The dot product measures how similar two vectors are in terms of direction.
  • Calculate it by multiplying corresponding components of the normal vectors and adding the products.
  • For example, with normal vectors \( \mathbf{n}_1 = (2, 2, -3) \) and \( \mathbf{n}_2 = (-10, -10, 15) \), their dot product is given by:
  • \( \mathbf{n}_1 \cdot \mathbf{n}_2 = (2)(-10) + (2)(-10) + (-3)(15) \).
  • This results in \( -20 - 20 + 45 = 5 \), which isn't zero.
Since the dot product isn't zero in our example, the planes aren't orthogonal. Understanding this concept simplifies the task of evaluating whether planes meet at right angles.

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

Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where \(K\) represents capital, \(L\) represents labor, and C and a are positive real numbers with \(0

Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x+y$$

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

The density of a thin circular plate of radius 2 is given by \(\rho(x, y)=4+x y .\) The edge of the plate is described by the parametric equations \(x=2 \cos t, y=2 \sin t,\) for \(0 \leq t \leq 2 \pi\) a. Find the rate of change of the density with respect to \(t\) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum?

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=x y \cos x y$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.